How difficult is the switch from flat to drop bar? I've ridden with flat bars for more than 10 years, both freeride and commuting. Now I'm considering switching to a road bike with a drop bar for commuting.

I went to a shop and tried a couple of such bikes, and the feeling was awful: a very bent neck and lack of control, esp. in changing the grip to brake. My question is: (1) what are the typical problems (or striking novelties) in moving to a drop bar, and (2) what can be done to alleviate them?

Some problems are highlighted in these useful discussions: neck, and wrists 1, 2. Is may seem that one should better not consider switching from flat to drop without a professional fitting (with an exception of this answer).
I have never had professional fitting - just adjusted until it felt right.  I am sure a professional fitting would be nice but I just did not want to spend my money on one. 

A road bike might not be the best bike for you.  In a drop bar consider cyclocoss or light touring.  Some companies have a drop bar "commuter".  They are going to have bigger tires and a bit more upright position.  If no other bike feels comfortable then stay with you flat bars.

As far as get used to it just ride. Brakes and gears are different but clearly you have figured that out.  More hand position.  Most riding is up on the brakes/hoods.  If you get on the bike and go straight down in the drops it will feel low - that is for high speed and into the wind.