I took off the cap that covers the whole in the sink because sometimes soap starts overflowing in the sink when i wash my hands. And since i took the cap off it doesnt happen anymore. But if i leave the top off can any bugs crawl through the pipe and into my bathroom?
Try pouring thick toilet bleach into the hole periodically so that the bugs don't come up
Don't worry about bugs coming up through the drain. There is a pee trap under the sink that seals the drain by staying fiull of drain water to prevent sewer gas (and any bugs) from escaping.
You're referring to the lavatory - the sink on your bathroom. Yeah, it depends on the size of the drain hole. I can't think though what would came from the pipes except for rats and vermin. Well, Sometimes roaches can hide from there. But not likely because of the trap.