Puzzle vs normal elo

I have a very high puzzle elo but a rather low normal elo. This is partially because I take my time and partially because I haven't played normally in a while. With my current puzzle elo, what rating should I be for other modes (bullet, blitz, etc)?

This is true for most players. There's no time or competitive pressure in puzzles. In general, there should be a correlation though.

Yes it's normal : rating puzzles are mostly higher than the elos of rapid eg

I think you can go to 1500

I have a similar puzzle rating but nowhere even near for anything else

Generally the slower I am, the better I play, but you cannot have 15-20 minutes per move in a game unfortunately

Looks like your ratings are almost in the right place. Maybe you are somewhat underrated for blitz like 100 points and 200 in rapid. In good circumstances you might peak at 1600 in rapid for a moment.

#5 you can in correspondence

Puzzles are also a lot easier because you know to look for a tactic which wins you an advantage, in real games you have treat each individual position/move as such which is a lot harder the average move is usually simple.

#7 I have thought about trying correspondence, but I don't know if it would work for me because I work a rotating shift, and with overtime I rarely go to bed at the same time more than a couple of days in a row. I guess I should give it a go.

Puzzle ratings mean absolutely nothing. The only thing that matters is ratings based on direct competition.