Removing old strings at the bridge
You are right to be cautious about pulling out a knotted string which is stuck in the bridge slot. Pulling it out using the long end causes the knot to rotate, and it might not have room to do so. Ideally (though it might not be possible) you should use long nose pliers on the shorter, "tail of the knot" end to pull it out. That way it should come out the way it went in.
Make sure you are pulling parallel to the front of the uke, not slightly upwards.Use a piece of thin cardboard, plastic, or something similar, to protect the front of the uke, as the pliers will be extremely close to the wood, and marks in that area are very visible.
Always leave a good tail on the knot when fitting new strings - not too long, just enough.
Hope this helps. Good luck!