Handicaps

I would like to offer my opponent a handicap of e.g. a rook.  Would it be possible to include a set of these handicap positions in the list of non-standard starting positions? 

Or is there some existing way of loading a board position and online-playing from there?

No, but I now have moved this to the Suggestion forum for you.

I also thought about this....but there is also another reason to offer this....in tournaments.

For example, I'm in a chess.com group that provides tournaments, but they randomly select the pairings.  Inevitably, a 1000 rated player gets paired with a 2000 rated player.

Has a chess rating handicap system ever been developed, even outside Chess.com?  Being able to setup a board, and play that board against an opponent, would provide a simple way to start this ("queens odds"), for example.