I read somewhere that you get exposed to a non-negligible amount of radiation from the sun when you travel by air. Would this affect an unborn child's fragile DNA?!
Unless you're flight crew, frequent flyer or there's a solar storm, radiation is not a significant problem. The radiation comes from space in general and not significantly from the sun, so a night flight has pretty much the same radiation level.
source: In-Flight Radiation Exposure During Pregnancy (PDF), American College of Obstetricians and Gynecologists.
This page contains the best Q&A about in-flight radiation concerns I've found. (Summary: It's generally safe.)
You can use this online calculator to determine how much radiation your flight would bring. To give you some perspective, this chart (here's another) shows that one flight across the USA brings you only 40 microsievert, which is one 25th of the yearly limit recommended by EPA, or just a few times more than a day's worth of background radiation on the ground. So if you fly 25 times in a year, you're still within safe limits.
Notable safety exception:
But even solar flares aren't that dangerous, according to the Q&A page linked above:
This radiation chart, while not made by scientists can help you get the gist of how much radiation can come from various sources. Notice that an airplane flight is in the first section. It has a lot of squares, which makes it looks scary. However, you'll notice in the second section that "living in a brick, stone, or concrete building for a year gives one almost twice the radiation that a plane trip does. And millions (billions?) of people live in concrete apartment buildings and have no problems producing healthy children. Hopefully this will help put you at ease.
Airlines will not allow you to fly after a certain amount, and this varies airline by airline. We've been turfed off a flight at 34 weeks, so I'd advise you to contact the airline before booking a flight anywhere; travel agents will not ask you if you're pregnant.
Some other points I have found:
This is a scientific study on the issue. I haven't read this cover to cover, but if you want actual facts and figures, I am sure they'd be in here.
But there is one paragraph in the summary worth note:
One final note (and this from myself) is that if you're carrying twins, I would be worried about travelling at any stage over 30 weeks as they are born earlier. Just something to keep in mind. I fly quite extensively and my wife and kids have always joined me wherever I am, whenever they can. She's flown many, many times pregnant, and as long as you take the precautions you need to (more exercise, more fluids etc), you'll be grand.
For reasonable amounts of air travel, radiation levels are too low to be a concern (per the Mayo Clinic):
Always check with your physician before flying while pregnant. Many doctors will tell expectant mothers not to fly after 36 weeks.
The risks of miscarriage or flight-related complications are lowest between weeks 14 and 28. During the first trimester morning sickness and other pregnancy symptoms can make flying particularly uncomfortable.