Is it possible for a wisdom teeth extraction site to close up while the stitches are still in there?

Yes: But the sutures are probably not in the socket, but rather outside the socket holding the gingiva together. Some sutures are resorbable, so even if covered by healing gum tissue, they will dissolve.

Depends on time: If the sutures are monofilament it is possible that the sutures can be 'buried' in the tissue. Usually it is time dependent. If there is no inflammation/infection around the sutures than over time it can submerge and be inside the tissue. That is why it is important to keep the postoperative appointment so the sutures can be removed if indicated.

Yes: With primary closure, the extraction site will often be closed immediately. Many surgeons use resorbable sutures that do not have to be removed, and will resorb over time.

Of course: The non-dissolvable stitches must be manually removed by the surgeon after 7 days.

Yes: yes, the stitches are there just to help attachment and to help close the area tighter as the initial healing is starting.