Is it necessary to include and update the status of a paper on arXiv once it has been accepted for publication in a journal? Let's suppose you write a paper and you post it on arXiv. If a paper is sent to a journal for peer-review, should you always include this information in the paper comments (including the name of the journal where the paper was sent)?

If the paper is accepted, is it necessary to include in the comments that the paper has been accepted for publication in... (and include the name of the journal)? Or it is better to wait until the paper is published and then include in the comments "Published in... [name of the journal]."?

I understand that it may take some for a paper to be published once accepted.
My practice is to replace the initial preprint with the postprint as soon as it is accepted.  After it is published (electronically), I go back and add the DOI and such.  The reason for this is that the paper is often significantly improved during the refereeing process, and I'd rather have people reading the improved (postprint) than the original preprint.