Project 64 2.x is known to run games at faster than their original framerate. If you want to emulate the game I'd suggest using either Project 64 1.6 or 1.7 as those are more console accurate than 2.0 or higher.
That can't explain the problems described. PJ64 2 is inaccurate, but it's hardly noticeable.
About the framerate: The game itself runs with 20fps. As far as I understand it, every (console) frame is shown three times by a TV, or in this case by the video plugin of PJ64 as that is the framerate of NTSC-systems (PAL would be 16.6/50fps).
Probably someone can explain that better than me, but in a nutshell: the framerate you see is correct and doesn't have to do with your problem.
Have you tried running other ROMs (OoT-J, MM-U)?