Chainsaw Repair

If you have the repair manual, there's probably no reason why you cannot run a screw into the pump and pry it out with a screw driver?

Likewise to drive the new one in with a soft hammer on a screw?

I would follow the repair manual for the depth.

My experience with these is that the scroll gear on the crankshaft typically uses little more than one thread on the crankshaft gear to turn the pump. To look at the crankshaft gear, it will look like the teeth are all OK but you need to look closely at one short section of the last thread on the gear in the rear.