I would like to know if it's possible to remove a cog thread from my face? (Photos)

Yes, we can remove those threads. I've done it several times for different people (which is why I don't put them in myself).
It can be done with local anesthesia right in the office typically.

Any theead will cause the inflamation, because we inserted something into the skin. We expect natural inflammations so your skin produce a new collagen. However, sometimes this process can be failed because of infection, pre, during or after the treatment. The best solution is back to the one who did yours because he or she knows better how they inserted the thread. Otherwise, plastic surgeon will do that too, but you need to get the information about the pattern they put the thread.