Should I warn my professor about some errors that I've found in his paper?
Ask him. It sounds like none of the errors is serious so if your question is just one of etiquette, all you need to say is that you think you found a few typos and little mistakes in the paper and ask if he'd like a list.

If the final version has already been sent to the publisher, it's probably too late to fix that, though there's the opportunity to correct small typos when reviewing the galley proofs. In many fields, people put their papers on their web page, too, so he'll probably appreciate the opportunity to fix that version, even if the version in the journal is already set in stone.