I will say, not necessarily, but you should check it and ensure you did not bump/move anything on the bridge. Unless it is a Floyd, then all bets are off.
You will most likely need to adjust the new pickup height.
On a Floyd Rose guitar it's actually easier and much, much quicker to swap pickups without removing the strings. Just unsolder, dive the Floyd, pull the pickup, squeeze the replacement pickup in, solder, adjust height. Need 1 or 2 rags to protect the pickups and the body finish. Black Winter bridge took a bit more patience since it's thicker and I had to be extra careful.
In most cases I would say no. Just will need to calibrate the pups to the guitar. Floyd Guitars are the best for pup changes IMHO. Just pop out the springs, pull the Floyd do your work pop the Floyd and springs back in and you should be in tune and good to go.
I concur - if nothing else is changed during the swap, no setup necessary.
If the new pickup is similar in output to the old one, try to set the height exactly the same to start.
on a tele bridge, maybe. depending how much the screws for the saddles got moved.
if you remove the batwing pickguard of an sg you should tape the height adjustment screws of the ABR/nashville in advance so they don’t change position
if you remove the batwing pickguard of an sg you should tape the height adjustment screws of the ABR/nashville in advance so they don’t change position