A trivial correction to my currently-under-review paper - What to do?
As you indicated, usually a paper is accepted with some suggestions - that is, the referee has a list of suggestions/errors. The author then addresses those suggestions/errors, and sends the edited manuscript back to the journal with a list of changes. This version is the one that officially gets accepted (I guess it's possible to go through multiple rounds of this, but I've never heard of this occurring in practice). 

Since the errors/omissions you found are minor, I would just wait to hear back from the referee, since they don't truly affect your paper in any significant way, and in fact the referee might find most of them. (If your paper is on the arxiv, you might consider updating it there). 

For what it's worth, you'll get the opportunity to correct typos even after the refereeing process; there is usually some editing for typos from the journal's side, and you'll also get a chance to make small changes to proofs before publication.