Can you activate an ability of a permanent that is currently targeted by a removal spell before the removal resolves?

Yes, you can absolutely activate abilities of a permanent that is targeted by a removal spell. Being the target of a spell grants no special restriction on the targeted permanent's controller. The removal spell sits on the stack waiting to resolve, and in the meantime both players can take actions.

This works because of the structure of the stack and priority rules (CR 116.1, CR 116.3). After an opponent casts a removal spell, priority passes back to you. You can respond by activating an ability of the targeted permanent — that ability goes on the stack above the removal spell and resolves first.

The only things that can prevent you from activating an ability are specific restrictions written on the permanent itself (like a 'can't be activated' clause), costs you can't pay, or timing rules (CR 602.2). Being targeted is none of those.

Concrete example: Your opponent casts Murder targeting your Llanowar Elves. Before Murder resolves, you activate the Elves' mana ability to produce one green mana. That mana ability resolves immediately (mana abilities don't use the stack per CR 605.3b), you now have the mana, and then Murder resolves and destroys the Elves.

This is a common play pattern — using a creature's ability 'in response' to removal squeezes value out of the permanent before it is destroyed. It is fully legal under the rules.