Our child is 7 days old. My parents live in another city which is a 6-7 hours travel away from us. As I'm working, no one is here to take care of my wife and kid. So we have to go to our parents' place to drop them both. Is it OK to travel this long and for my wife to have the kid in her lap? Seeking experts' advice.
Zero distance in someone’s lap. In an accident baby will become a missile and escape the grip...
If the baby is in a properly installed, rated car seat then fine, otherwise how would many parents get babies out of hospital?
Make the baby comfortable, have sufficient food, stop for breaks as necessary. Natural is easiest... otherwise you need warm water somehow - thermos flask or means of heating or a bottle heater which keeps the bottles warm...
We found that once fed, babies would sleep if you drove smoothly, but if you are rough they wake up, and they can sleep for 2, 3 or 4 hours solid in the car - done it often.
As mentioned, a baby should never travel in the car by anything but a child seat that is properly installed (preferably rear-facing). Do not take them out of the seat for any reason while driving: Not to feed, not to console them, not for anything but extreme situations. Always pull over first or find a parking spot. Accidents only require a fraction of a second.
Next to that the NHS advises that The Royal Society for the Prevention of Accidents recommends taking at least a 15-minute break every two hours . Multiple other sources support this "two-hour rule", stating that prolonged travel may lead to breathing and spine issues due to being in an upright position. , , 
Research seems to agree that the less time spent in a car seat for a child that age the better (some sources going so far as to claim that two hours a day should be a maximum), it is understandable that this is not always an option. It is recommended to keep the travel time to an absolute minimum where possible.
Another option would be for you to invest in a lie flat car seat  (though their safety is disputed, see the comments). Make sure it confirms to all child safety guidelines. These would allow for longer travel per day (The two-hour rule still applies) because of a more comfortable positioning. Or, ideally, persuade your parents to come to you!
Make sure that whatever you decide is in the best interest of both the baby and its mother. An infant is a fragile thing, its safety and comfort is more important than that of any other parties involved.
Travelling with newborns is not the safest option, but it can be done:
- Carrying the baby in your wife's (or anybody else's) lap should be avoided at all costs. If you e.g. brake just a bit harder than usual, the baby could slip from the lap and get hit in the head etc. Get yourself a proper baby seat. It doesn't need to be expensive or new, but it needs to be in good condition (no cracks, no broken straps).
- Do. Not. Hurry. Accept that it's going to be a long trip. If your wife wants to breastfeed, stop the car. If you have to burp the baby, stop the car. If a diaper needs to be changed, stop the car. If the baby is crying and you just need to hold it for a minute, stop the car. If you need to take a break because the baby's taking on your nerves, definitely stop the car. Do not take the baby out of the seat while the car is moving, ever.
- Make sure the car's air conditioning system works. Newborns are very sensitive to temperature, and they can die from a heat stroke on their sleep if the car gets hot, even while you are driving. You don't have to keep the a/c running all the time, as that might make the air too dry for the baby; just use it as needed to keep a comfortable temperature inside the car.
- A moving car is like a cradle for newborns: the smooth movement and the engine rumbling may make them sleep longer than usual. But the baby still needs to be fed every 3-4 hours at least to avoid malnutrition, even if they are asleep. So if you've been driving for long and the baby is still sleeping and you think you can just keep driving and make the trip shorter: don't.
If you follow this advice, you should be able to make the trip safely and without problem.
The above being said, what is the safest option, then?
Well, if I were you I would ask your parents to come over to your place to help your wife. I understand there might be cultural issues in doing this, but it is still your best option. That'd be orders of magnitude better than your wife and 7-day-old newborn enduring a 10-hour road trip.
Some newborns may be sleepy and not interested in feeding. At first, babies need to eat about every 2 to 4 hours to help them get enough nutrition and to grow. This means you may need to wake your baby to feed. You can try patting, stroking, undressing, or changing the diaper to help wake your baby to feed.
Newborns will wake up and want to be fed about every 3-4 hours at first. Do not let your newborn sleep longer than 5 hours at a time in the first 5-6 weeks.
Source: Cleveland Clinic
Definitely not ok in someone's lap. That is incredibly risky.
Even in a proper car seat, if you can borrow or hire one, a journey of 6 hours may damage the baby's back. Make sure you take regular breaks that allow the baby to get out of the seat for some time.
Please, try to get the parents to come to you.
Some of the information contained in this post requires additional references. Please edit to add citations to reliable sources that support the assertions made here. Unsourced material may be disputed or deleted.
Because of the cultural differences and subsequent (very strong) opinions about traveling without a car seat, As I warned you in a comment, you'll not receive an answer that states it's safe. Knowing that the infant is safest in a car seat, I hope to give you some advice that will actually help you in the event that it is not possible to travel in this manner.
The other answers are full of good advice. Try to find/buy/borrow/rent an appropriate car seat and take the precautions mentioned (rest periods, etc.)
If you absolutely cannot get a car seat and must travel, use more than common sense. Some things I would recommend:
Travel when and where there is the least amount of traffic. Do this to minimize the odds of an accident.
Travel well rested. (Do this to minimize the odds of an accident.) Your wife should also be well rested so that she can remain vigilant.
Have your wife sit in the middle of the back seat, belted in (best with both lap and shoulder restraint.) According to both the National Highway Traffic Safety Administration (NHTSA) and the American Academy of Pediatrics (AAP), the most dangerous spots in the car* are 1) the front passenger seat, 2) the driver's seat, and 3) the rear side passenger/window seats. The safest seat is the back seat in the middle. (If your car doesn't have rear seat belts, please borrow or rent one that does. You may even be able to rent a car seat.)
Do not belt the baby to your wife. If she is thrown forward, she will injure the baby badly. If she can swaddle the baby securely to her upper chest area, this is safe and wise.
Have some pillows to place on her knees behind the baby and on the floor, such that if the car stops so suddenly that the baby flies out of her arms/away from her, it will hit the pillow(s) and fall into the pillows. The most common accidents are front end and rear end collisions. The baby's injuries should be minimized (but not necessarily prevented) if these precautions are followed. This does not hold true in the event of a side collision.
Regardless of the wisdom of using car seats, in the US people do sometimes travel (e.g. in taxis or subways) without them. It is not even illegal in many cities. But you will not hear anyone to admitting that.
*Provided all seats, including the rear middle, have three-point restraints. If the middle does not have a three-point restraint seat belt, choose a back seat that does.
Adults with certain medical conditions also must take precautions. Adults over 40, obese, and others risk blood clotting and pulmonary embolisms on long drives.
Seems like all other answers have addressed a child safety seat issue (which as someone who grew up outside of the First world and yet lived I find questionable), but I want to focus a bit on other aspects of the car travel.
Car travel differs from home stay by several factors:
- the air your baby will breathe
- the temperature might be different
- noise level might be different
- motion and vibration
In many places around the world, the air will sometimes be dirtier along the roads, and also if your car is in a bad shape, it might leak some exhaust gas into the passenger area. The carbon monoxide in this air prevents blood from carrying the oxygen properly, by reacting with hemoglobin. This is reversible (if exposed to a relatively low amounts).
This site (https://www.childrenscolorado.org/conditions-and-advice/parenting/parenting-articles/prevent-carbon-monoxide-poisoning/) mentions that "Young children are especially vulnerable to the effects of CO. They are more susceptible to carbon monoxide and may experience symptoms sooner than a healthy adult." Be careful, but don't be too paranoid. In industrial areas of China babies are still born and grow even though the air is polluted as nowhere else.
The second thing is temperature. Most likely it's gonna be heat if your car does not have an air conditioning. This is fine, you just have to make sure that the baby's clothes are adequate.
Car noises should not be an issue. This is quite obvious, but I included it for consistency.
As for motion, I could not find any articles on damages to vestibular system development caused by prolonged riding in the car and acceleration/deceleration that is involved. This also makes sense, as plenty of babies were born on ships or have sailed with their parents, loading their vestibular system much more when traveling by car, and stayed healthy.
All in all, I think that it boils down to your best judgement on: would this car ride impact a health of any person, and if the answer is no, then your baby should be fine too.
Good luck and I wish a good health to your little one!
Safety issues have been thoroughly covered by other posts, so please read those first. But on a personal level you will want to stop every 2 hours at most and change/stretch the baby because car seats tend to cause diaper blowouts (poop leaks out the top and sides of the diaper) due to the baby being pressed butt down in a tight seat nook for so long.
Even if there is only a little pee in the diaper I recommend keeping the baby in a dry diaper the entire ride to give allow for maximum wet poop absorption and minimize the chance of a leak. This will make sure the baby is not sitting in poop that has leaked out which can get on their hands and mouth.
Congratulations on your new child you are now the proud parent of an organism that is the product of 3000 million years of ongoing development.
I know that in a world infested with virtual reality we can easily lose sight of the actual reality but your baby is a remarkably robust creation.
I do however understand and acknowledge a significant number of parents today seem reluctant to acknowledge that there are no 100% guarantee in life and life remains, albeit significantly reduced, a risk rated adventure.
I do not know in which country you will be making your journey and I can only speak of the statistics for the country where I live but one person is killed for every 120,000,000 km travelled so statistically you have more chance of winning a big lottery than being involved in an accident in your journey.
The take-home message is don’t get things out of perspective.
If you were flying your baby would travel safely on one or both of its parent’s chest strapped to their body. Some (we are still all different) are mildly affected by the pressure system of the cabin but that aside most travel comfortably in that environment for 2 to 3 times the period of your journey.
Your baby will be absolutely comfortable within an environment in which you are comfortable.
Babies require constant food and water as long as you continue to deliver both these products the rest is up to random numbers.