Yes Premiere pro can handle different frame rates.
In terms of what frame rate you should use for your timeline, I'd (generally) go for the one that matches most of your clips.
29.97 and 59.94 are multiples of each other so either of these frame rates would be fine for the 59.94 & 29.97 footage you have. However I would likely choose 29.97.
For the 25fps material I would drop it into your timeline and apply speed/duration + optical flow to each of these clips -
> right click a clip in your sequence and select Speed/Duration > then select Optical Flow from the pull down 'Time Interpolation' menu.
This will (in most cases) create a very good standards conversion of the 25fps footage to 29.97. You may see some weird artifacts on 'complex moving shots' (and some other specific circumstances) but often it will look great. You will need to 'preview render' your sequence to see how the optical flow looks. For problem shots turn off optical flow or select 'Frame Blending' from the pull down menu mentioned above.
As to your audio sync issues - this may not have anything to do with the different frame rates. Do you have any 'variable frame rate' clips in your sequence. i.e. shot with an iPhone etc?