Do publishers do a punctuation check or reject books that could have mistakes? Do they edit a final version?

Here's the deal.

Publishers are looking for writers. That is, people who have good stories to tell, and have mastered the English language enough to do that effectively. Time is money, and there are enough good writers submitting things that they do not have to waste their time cleaning up bad English.

Have a few errors here and there? Not a problem. The manuscript will be edited several times in the publishing process and those mistakes will be caught. Everyone, even bestselling authors, makes a mistake sometimes.

Have a lot of errors? Bad grammar, run-on sentences, poor punctuation? You run the risk that a publisher will feel you aren't a good enough writer to be worth his time. And at the point he decides that, he will stop reading. So if you are trying to sell your first book and don't think your English is good enough to pass muster, pay for a copyeditor to look at it before you turn it in.

Please note that spelling errors are looked on with especial disfavor, since ten minutes with a spellchecker can correct an entire manuscript. When a publisher sees a lot of spelling errors, it tells him you didn't care enough to do that.