I want to know if a cursed hero can still use abilities. The FAQ says yes, but for example could runemaster thorn use his ability to teleport or could one fist do a unarmed attack?

bomber said:

It is not the 'use of abilities' that is allowed or prohibited. The monkey has specific restrictions. Follow those restrictions to figure out whether the individual abilities can be used or not. Some will be usable, some will not.

DJitD pg 16
Monkeys cannot attack and cannot use any items, including potions. A monkey can move up to 5 spaces on the hero’s turn, but cannot take any movement actions

FAQ pg 4
Q: A hero who has been Transformed by either Curse of the Monkey God ... Can that hero take other actions, such as receiving an order token or declaring a Run action? ...
A: A Transformed hero may not declare any action . ...

Thorn's ability does not require any declared Action, is not a Movement Action, and is not an Attack. There is evidence to suggest that a monkey actually gets 5 MP (which still cannot be used for Movement Actions) rather than moves 5 spaces - a monkey presumably should still be slowed by terrain, climbing out of pits, trapped by webs, etc, and Talia's ability which allows her to 'move spaces' has been clarified as giving her MP instead. Thus, monkey-Thorn could spend his 5MP (remembering he can also get MP from spending fatigue) to use his ability.
Monkey-One Fist could not use his ability as it is an Attack, something monkeys can't do.
SImilarly, as examples, monkey-Steelhorns could not use his ability (it requires a Run Action, which he cannot declare, and it is an Attack, which monkeys can't do), monkey-Lyssa could, monkey-Trenloe could not.
Monkey-Zyla would still have Fly and Ghost, monkey-Sahla could spend fatigue to remove effect tokens (are the monkey 'wound' tokens effect tokens???) and monkey-Grey Kerr could not change his declared action because he doesn't get to declare an action in the first place.
Monkey-Talia, if given a Guard order by Leadership, could use it to move with 5 MP (or maybe only her normal move?), though not to actually attack.
Monkey Silhouette, if given a dodge order by another hero's Leadership, could spend 2 fatigue at the start of her turn to change it to a Rest and recover all her fatigue.
A monkey with Leadership could not use Leadership at all, as it requires the declaration of a Ready Action.