Okay, here goes:
Various drums: I like to keep all the percussion on a 5 line staff. Standard form would be to put snare drum on the C space (if this were treble clef), bass drum as F, and cymbals if you have them in G above the staff, with an x for a notehead. All this depends on how many parts you want on the same staff, and you have a lot of freedom as long as you put some text at the beginning of the piece to define which instrument reads which note.
"Tree bells": I'm not sure if by this you mean a mark tree (metal cylinders hanging in a line from long to short) or a bell tree (cup chimes stacked on top of each other). Either way, the notation is a note at the bottom of the staff such as E or F with a glissando line up (without an end note), or if you want the pitch to descend instead of ascend, starting at the top of the staff with F or E and a glissando down.
Timpani: As pitched percussion, timpani are pretty simple - just write normally in bass clef. Keep in mind that your timpanist can only have 4 (or maybe 5) notes available at once. You also might want to look up standard tuning ranges of timpani to make sure that they can handle the distribution you want. It can be helpful to notate the tuning changes (eg Ab/C), but it's not necessary. In fact if you really want you can just write the notes you want and let the timpanist figure out how to tune - which could turn out fine or near impossible to play. I suggest you have a timpanist look over the specific part.
Rain sticks: I've never seen notation for these, but due to their nature I would notate them just as I described bell trees and mark trees earlier, except in this case you can't really make the pitch ascend or descend, so just pick one.
Good luck.
