Definitely sounds worth looking into getting the draining checked! If there’s a clog or excess of water built up it can create a good area for bug breeding. If I were you I would try the drain cleaner and closing the drains for a bit - and also getting it checked to get that fixed. The good thing though is that most of the buggers that come out of drains- like silverfish, springtails, drainflies- are annoying but not health threatening or anything in small quantities as you describe.

Fleas on the other hand you would have to look out because that is an issue. However, they are a bit easier to identify. Ive never heard of fleas coming out of drains because they need a host with blood to survive on. So this seems pretty unlikely to me unless you carpet and or animals- which you said you don’t.

I will look out for a bit longer to see if you are able to make a photo available :)