Is there a separate web app that supports this? Or at least a place where I can set up a digital board to freely practice on.
You can freely test moves on an empty board on OGS by clicking the upper left logo and selecting demo board.
If you do not want/need it stored anywhere and just want to click stones around without much further tools, there is bunch of online services, try https://goban.co/ or google.
To copy a game state, that is more tricky, you can just use the analysis tool or make a review, if you want/need it completely separate you would need to download the .sgf first (from the right hand game menu) and then upload that to your library (upper left OGS logo > library)
Yes, and many ways to do this!
It sounds like you want to “start a review”. That bar on the side (or on the bottom in mobile) has icons. The circular arrows is for starting a review. ( https://github.com/online-go/online-go.com/wiki/Reviews,-Demos-and-Puzzles#review-an-existing-game)
If you want to review outside of OGS, there is also the option to download an SGF.