When the sink is on in the half-bath, the toilet fills up with water and does not flush. After several minutes, it will drain by itself and then bubble.

Your drain downstream of both fixtures is clogged. You can try to rod down the drain that goes straight into the wall after the sink trap or you can pull the toilet, the toilet stand a better chance but involves a bit move work

Depending on the system layout, you have a partial blockage in the drain line. If the sink is after the toilet, it is blocked after the sink and is backing up into the toilet. If the toilet is after the sink, the blockage is after the toilet. You could try plunging but will probably have to get the drain line snaked.

Thanks a lot guys. I will call the plumber and get him over here tomorrow morning. Have a great evening.