Are we supposed to put the new diaper on immediately or give our newborn some diaper-free time? His skin under the diaper is slightly moist when changing it, should we give him some time to air out? If so, how much? And what are the negative consequences of just putting a new diaper back on him?
Most of the time it's okay to just put the clean diaper right back on. If it's too moist too often, he will get a rash, but usually that's more due to waiting too long to change rather than putting a new diaper on too soon. Make sure to clean all the urine off so the moisture is just from the wipe. You can use some baby powder if it starts to bother him, and there's a gazillion ointments out there too.
That being said, there's no down side to diaper-free time, except to your carpet!
How much time to air:
After cleaning, you can dab off the skin with a soft towel (paper is likely to irritate more than cloth). If the skin still feels a little moist, it takes only 30 seconds to air-dry, and then you can put the new diaper on. You can also softly pat the skin with the back of your hand to speed up the air-drying.
Especially with newborns, there's often no need for soaps, ointments or powders as long as the skin is clean and dry. We use only warm water, and cosmetic pads rather than wipes. We do use an ointment on occasion when the skin is red, otherwise nothing.
Diaper-free time is good. We used to put our newborn on a soft blanket to air out and "play" for a little while before dressing her up for bed.
The only caveat is sensitive skin. We always had some sort of layer between her and carpet, simply because we have 2 dogs and 2 cats and didn't want any additional skin irritation to happen.
The skin should be dry quite quickly, so normally there should be no problem to replace the diapers instantly with new ones. You can use a soft towel and the remaining moisture also should be absorbed by the diaper. Just if you see that the skin is or gets red or irritated, this might be a sign to give it more time to "breathe" and air out.
I'd also agree to the suggestions above: warm (pure) water and a towel should be enough to clean.
Further benefits of giving some diaper-free time (even if there are no "bad" signs like irritated skin):
I think it's good to let your child also play naked, because it can only move freely if it's not hindered by the big diapers.
And you can use this time without diapers for cuddling (maybe under a blanket, if it is too cold) and if you take your shirt off yourself, the baby also gets some contact skin so skin, which is also considered important for the relationship to the parents.
... so you could combine business with pleasure. ;-)
In my experience, slightly moist is okay (e.g. moist from a diaper wipe). That said, once or twice a day, I put a diaper under her bottom and play peek-a-boo with her for a few minutes on the changing table. (Do be sure to strap the baby in of course if you're going to duck under the table!) I don't feel this hurts, although it's hard to say if it helps do anything other than make diaper changing times more fun for her.
For my baby, diaper rash is more a function of teething and her dairy allergy. We change her frequently so we haven't really run into diaper rash apart from the teething and the food allergy. However, for the teething rash, her ped did recommend letting her air out a bit longer than we otherwise would. Nothing really life-changing, just a few minutes between taking off the old diaper and putting on the new one.