When I first saw this I thought I was imagining having seen it before but it turns out there is a similar discussion going on
To answer your other questions. Yes it is fine to be with other registrars and it's recommended, as depending on how you sell your domains you may need to push* a domain instead of transfer it out. (*move to another account within the same registrar)
As for the ones to avoid, a search for the name on here, Google etc, or an ask on here should be able to inform enough as to whether they are trustworthy or not.
The popular ones off the top of my head are:
GoDaddy
Namesilo
Namecheap
Dynadot
Uniregistry
The more knowledgeable and

will be able to suggest many more.