I'm stuck at the point where a bouncing tomato (I guess it's called Tomatl) goes in circles through the black thingy (Sadista?) dead zone. A tree nearby tells me that Tomatls can make me invisible to Sadistas if they are big enough, but I can't figure out what to do exactly.

You'll need to activate one of the tiles that takes you to the future, and you have to time it so that the Tomatl is in a position between you and the Sadista. This forest has a lot of such puzzles.
In the 'present' I believe you can freeze the Tomatl and it is enough to be able to block the Sadista's attack.

No, you can't do that, Tomatl will automatically move to predestined position in the future.
I've figured out what I was missing. All I had to do - push those red and blue mushrooms with Menos' charged attack. I've successfully found key fragment and met Nawi, but anyways your advise was highly appreciated.

Yeah that forest blows your mind. Took me a while before I got everything figured out AND working.