How does the multiplayer in Hill Climb Racing 2 actually work?

You're racing against downloaded copies of other people trying to do the run.

In order to prevent unfortunate lag, or unfortunate lagging, (as well as allowing you to pause the game), the other players have their attempts of the map downloaded onto the game before being played, like ghosts.

I couldn't find any information to confirm or deny if it's a player's "best" times, however given how frequently I come across other players who actually don't make it to the end, as well as the fact that often (in the higher leagues) it takes longer to find a match, I do believe you are playing against players who are online and playing that cup at that given moment (or have done so very recently).

To answer if you're also in those matches - I presume not, as I've never seen someone time out mid-match, or leave mid-match, so presumably once you've completed a cup your go is uploaded and can be used for the next x minutes to play against other players.