My two cents...
X513, Lots of questions in a small space so here goes:
Question: When should I be in the drops, on the hoods, bars, etc.? That is, where are best positions for climbing, downhill, etc.?
Answer: Do what feels good. However, make sure that you are on the hoods or drops if you think that you might need to brake. That said, most people prefer the hoods when climbing out of the saddle. Some rare hammerheads will use the drops. Climbing in the saddle, most folks use the tops or hoods. On descents, most people favor the drops for better aerodynamics, bike handling and braking control.
Question: Brake from the drops or hoods - both?
Answer: Both, but use the drops in situations that require hard braking. This will give you more control over the bike and it is easier to exert the required pressure on the brake levers.
Question: I seem really bent over in the drops, maybe I should flip my stem around so I get a little more bar height?
Answer: Position on the bike gets discussed endlessly in this and other RBR forums. For most people, especially new riders, position on the bike evolves over time. Initially, the goal is to be comfortable on the bike so you can get used to it. After, that it is a process of refinement.
Given that you are new to road biking, IMHO if you are not comfortable, yes flip the stem. This will provide a quick and dirty fix. As you get more comfortable, you may want to go to a shop that offers fitting services and pay to be “professionally” fitted. Also, do some searching here on RBR. There is a wealth of information on “proper” position on the bike.
Hope that this helps…