Good final work!
Is it best practice to try to maximize the use of the tileset?
I'd say it's best practice to use the fewest number of tiles possible to get the message across. That is to say don't try to force everything to be 256 tiles if it doesn't need it because you can save space that way, but don't feel bad about using 256 tiles if that's what your art requires.
Are there RGB values listed somewhere for NES colors, or will it be fine to screengrab the colors from the NES Screen Tool and eyedrop them?
Believe it or not, there's no such thing.
Your colors are going to look different on everything, and that's just the way it is. Edit: I realize why you asked! Screen grabbing NES Screen Tool is fine if you're looking for a palette to use for future projects.
edit2: I personally like this palette:
which is the top four rows generated here: https://bisqwit.iki.fi/utils/nespalette.php
(It may have been addressed already) Is there a way to reuse tiles but V or H flip them on the nametable?
Nope. You don't benefit tile count-wise from being symmetrical for the background.