the pop up , where you pull up on the lever so the sink can drain and fill, these get full of hair, soap scum and tooth paste and clog up over time. Thats why the pee trap is fine and thats why the pipe in the wall is fine. The clog is in between the pee trap and where the water drains into the pee trap. we call it the pop up. you can try using a piece of wire and try and fish out the stuff stuck in the pop up , or you can remove it and clean it out , but this is where the problem is coming from after the information you have given to me.
No. This sink is not located near the a/c.
It did before the water started coming up through the drain. We haven't tried to use the sink since this started. We have to bail out the sink every few hours so that it doesn't run over.