Wood is so popular as a building material because it is both strong yet flexible, and as a bonus is easy to work with.
Minor bends and warps will no matter in the end.
1x1 inch is enough for the "blocking pieces" that you need. Strong enough and big enough to take screws from two directions.
Power tools? All you really need is power (hand) saw, jigsaw, drill. Table saw optional if you can't do a straight cut otherwise (but table saws are statistically the most dangerous of all power tools, seconded by the hand power saw mentioned earlier). Basic router if you want T-molding (good idea to do this) and a few other fancy things. You won't need a jointer or planer if you are just buying wood pieces from hardware store. Get at least one pair of G-clamps (I have small, medium and big) and a larger pair of F-clamps, these guys are so useful you won't know how you managed without them.