You may choose to remove Sarulf's +1/+1 counters even if that would cause lethal damage to be marked on Sarulf. If you do this, Sarulf will die after its last ability is finished resolving. But it won't have gone alone! Legendary.

You choose whether to remove +1/+1 counters as the last ability is resolving. If you do, the appropriate permanents are then exiled. No one can respond to your choice or take actions during this process.

Choosing to remove all the +1/+1 counters on Sarulf won't affect any other counters it may have.

If you do remove +1/+1 counters, you must remove all of them. You can't fine-tune the carnage by only removing some of them.

It doesn't matter which graveyard the permanent is put into as long as an opponent controlled it when it last existed on the battlefield.

If Sarulf isn't on the battlefield as its last ability starts to resolve, you can't remove any +1/+1 counters from it, even if it had +1/+1 counters when it was last on the battlefield. No other permanents will be exiled.

If Sarulf doesn't have any +1/+1 counters on it as your upkeep begins, its last ability won't trigger at all. If the ability does trigger, it will check again as it tries to resolve. If Sarulf doesn't have any +1/+1 counters on it at that time, the ability won't do anything. No other permanents will be exiled.