Pain in the lower back and neck during cycling
Your bike fit should not cause you pain. Back pain, neck pain, knee pain, hand pain, elbow pain, etc are all signs of a problem; they shouldn't simply be ignored, and probably won't just go away on their own. Muscular soreness after pushing extra hard is really the only kind of pain you should get.

I suggest that you:


Get the bike fit to you. Many bike shops offer this as a service, and may discount it significantly with a new bike purchase. This will probably entail assessing your flexibility and changing out the stem and adjusting the seat position.
Increase your flexibility. Hamstring stretches, etc, so that you can bend at the hips more. Personally, I do this while coasting on the bike after I've warmed up.
Strengthen your core. Planks, etc. So you can hold that position better and longer. As long as you make an effort not to lean heavily on your hands, this should improve as you ride more.
You may also want a different helmet and/or sunglasses. Anything that blocks your vision when looking "up" will tend to force your head to tilt back more. You don't want a visor/peak on your helmet when riding "down low" on a road bike, and you may find that thick rims on the top of sunglasses are also a problem.


From your description ("when [...] in a low position"), I'm guessing that you're putting yourself in a lower position than you're flexible enough to really do right, and compensating for that lack of flexibility by curving your lower back too much and bending your neck back.

There's a lot of debate about ideal bike fit, but in general you want to be:


Bent mostly at the hips, with a bit of spine curvature (not too much).
Your upper back tilted back so that your neck doesn't have to bend too far back, which is facilitated by less lower-back curvature.
Shoulders down/forward and elbows slightly bent. Not leaning too heavily on your hands. You should be able to take your weight off your hands for a bit without shifting your position.


On a road bike, you'll typically spend most of your time with your hands on top of the drops, so that's the position to make the most sure you're comfortable in. But you want to be able to be reasonably comfortable down in a lower handlebar position ("in the drops") for windy situations, downhill sprints, etc.