10 gallon is fine for your baby and will be big enough for many months. Starting out with a 20 gal is also usually possible, but young BPs can be overwhelmed by all that space, so clutter it up with extra hides or fake greenery. Some BPs would be stressed by even a 10 gallon and do better in a 6 qt shoebox sized plastic tub. You'll just have to figure out what works for you & your snake.

As far as your heating questions, yes you can control a hot side and a cool side with one t-stat, as addressed in the previous posts. The best way for you to do this will depend on the conditions in your house. You may be ok with just one UTH and a well insulated tank. If that does not keep the cool side warm enough, then play around with it until you find out what works for your situation.

Lamps do a better job and UTHs at raising the ambient temp, but the trade off is they can cause humidity issues. If you keep your house somewhat cool, it may be easier to overcome the humidity problems than it is to get the tank warm enough using only UTH.

It is very good you are going to get your enclosure set up ahead of time, because there is no "one size fits all" answer as to how to balance the proper temps and humidity in a tank, so most people have to do some tweaking before they get it right.