DOIs are cheap. Not having them is a definite red flag.
Being OA with <5 papers means, at least, this is a new journal. Depending on how many years it's been since the first paper was published, it could mean the journal's editorial board is inactive, or the publisher is not giving the journal much attention. If the journal does not go anywhere, or (even worse) if it shuts down, your paper is not likely to be read much.
Finally you write that a journal emailed asking your professor to submit the paper to them. This is weird, because the journal is not likely to know your paper's been rejected by some other journal.
This doesn't mean you shouldn't submit to the journal anyway, but be careful.