Hey,
They are probably less inclined to remove it in case they disturb any clotting process underneath. It won't do you any harm to leave it in fortunately.

On a completely non-professional note and on a personal level, I had wisdom teeth and molar teeth out for orthodontics and would very gently swab the socket with a Q tip soaked in salt water. Definitely don't put a dry one in there. You can do that to remove any plaque or general gunk around the socket. The alvogyl may come out but if it comes out then it was ready to come out anyway if that makes sense.

As far as the squishiness goes its just your body increasing the fluid flow to the tissues to promote healing and repair, it'll do the job on its own. If there is pus you should be able to see it. You have had quite a lot of antibiotics though and probably won't need any more unless really necessary.

For pain just continue taking any painkillers given, or take whatever you would take for a headache.