Wouldn't call this a rhytm game. It's more of a "click before the bar runs out". Honestly just clicking very often does the job.
And when i actually try to click to the rhytm it doesn't seem to work.
I agree. I tried only letting the snake move on the beat, but that made the game very slow. So the current idea is you need to move at least once per beat, but moving several times is allowed.
I am also not sure how to improve the second issue. I would like the timing to be intuitive for new players and it doesn't sound like it is. Any suggestions?
Depends on how you coded the timing. It would be nice to have like these threshholds where you can click only once. Or the bar that is shrinking to the middle, you should be able to click between when the bar is fourth of its size and when its a single line. And you should be able to click only once then. That way players cant spam their keyboards like i did.