"Have there been any studies into how frequently players are able to correctly predict the opponents next move? If so, how does this vary across the different rating levels?"
Even grandmasters fail miserably at predicting the next move of their fellow grandmasters. In most positions there are two or more viable candidate moves. Carlsen said he usually calculates 3 possible responses.
What is a good rate for predicting the opponents next move?

It's an interesting question, and I don't know the answer, but I want to maybe change it a little. I don't think it's important that you predict your opponent's next move as much as you want to have considered the possibility. If they play a move you didn't consider at all (and that move is pretty good... not necessarily winning but at least pretty good) then you screwed up your analysis.
I mean... sometimes there are certain types of moves. Such as "type A I will do ____ and if type B I will do ____" and that's fine, but again, if they play type C, and it's good, and you didn't even consider that possibility, then you screwed up.

I saw an old interview of Bobby Fischer shortly after he won the World Championship where he was asked about long calculations and he said it depends on the position, in some positions there is only one line so you don't need to think about other responses, and in other positions there are four good responses on the next move and the move after that.

Yeah, and if there aren't many forcing moves, then it makes no sense to calculate a very deep line.
So for depth and width, it just depends.
And also on time control... obviously for blitz or similar, you're not going to spend a lot of time considering things.

"Have there been any studies into how frequently players are able to correctly predict the opponents next move? If so, how does this vary across the different rating levels?"
Even grandmasters fail miserably at predicting the next move of their fellow grandmasters. In most positions there are two or more viable candidate moves. Carlsen said he usually calculates 3 possible responses.
I think at that level it actually becomes easier to predict your opponents moves because the top tier GM’s are constantly playing against each other over and over again
The thinking process for choosing best move involves thinking about how your opponent might respond and how you would respond to that.
I think I have gotten slightly better at this over the past few months but I still sometimes surprised by the moves my opponents follow up with. Generally I am not surprised by tactical moves as I try to look out for those, but there are often strategic moves that change the complexion of the game that I did not anticipate.
Have there been any studies into how frequently players are able to correctly predict the opponents next move? If so, how does this vary across the different rating levels?
Don't worry about predicting your opponents moves, worry about seeing good moves for your opponent and thinking how to counter them. That's way more important. On this website you can watch other people's games and predict next move. I did it for fun one day, I was watching a lot of games and predicting their next move. I could predict moves of players in my rating range (1600-1800 blitz and about 1900 rapid) way better than I was able to predict the moves of below 1000 players.
Also GMs rarely play weak moves, so the number of plausible continuations is much more limited.
I think you missed the point Fischer was making. When he said there is only 1 continuation to look at, that didn't mean I will calculate this, because this is what my opponent is going to play, but because this is the only thing that should be calculated. It will be a lot easier if my opponent does not play that move. When I am thinking on my opponent's move, I am thinking about responses to moves which I find good. If my opponent plays a move I didn't see and I think it is a bad move, then I am glad he played such a move instead of the ones I was calculating.
The thinking process for choosing best move involves thinking about how your opponent might respond and how you would respond to that.
I think I have gotten slightly better at this over the past few months but I still sometimes surprised by the moves my opponents follow up with. Generally I am not surprised by tactical moves as I try to look out for those, but there are often strategic moves that change the complexion of the game that I did not anticipate.
Have there been any studies into how frequently players are able to correctly predict the opponents next move? If so, how does this vary across the different rating levels?