A paper is initially posted on the arxiv when both collaborators agree they believe it is correct and written well enough to submit for publication. Often a submission for publication directly refers to the arxiv posting rather than having an attached paper. This then becomes v1. I will admit I have occasionally posted papers I was not ready to submit, in order to establish precedent. In this case the theorems' proofs were complete but perhaps a background section or better exposition was added before posting. Be sure all coauthors are given ownership of the paper. Some mathematicians are under intense pressure to post on the arxiv either because they are facing the job market or because they are being reviewed for tenure. Coauthors should be considerate and work together to complete a paper in a timely manner. But always remember that v1 of every paper on the arxiv is permanently on the arxiv even as later versions are added and even if the paper is later withdrawn.   So it should be correct.

Once the paper is refereed, usually a few minor errors have been found and the paper is either resubmitted to the same journal or another one. When the necessary changes are made, the paper may be reposted as version 2 on the arxiv. While it is rarely done, a second version can have a new name. One does not withdraw a paper, one just posts a second version that is correct. One gives an indication of the changes. If things are very simple a single coauthor can handle this task as well as the resubmission. If a serious error has been found, that is another story which will be discussed below. If a second referee (at the same journal or another one) requires additional changes, another version may be posted on the arxiv when it is resubmitted again.

Once a paper has been accepted, one of the coauthors can go on the arxiv and add "to appear in ---" . Once the proofs have been received, the full journal reference can be added. I also like to repost on the arxiv if the journal has renumbered the theorems and/or equations. This way people who do not have access to the journal can read the preprint instead of the paper and can quickly go back and forth between the two. Some journals allow the actual published version to appear on the arxiv bot most journals will not allow this.

If there is a major error discovered in the paper before publication, there is a serious choice which must be made involving all the coauthors. Taking a few weeks to try to resolve the problem is acceptable. If it takes longer, one is expected to withdraw the paper from the arxiv. The withdrawal should say explicitly what is wrong in the withdrawal. If only one theorem or one section is incorrect, then you can post a new version of the paper with that theorem/section removed and write a comment explaining that this was done. This is preferable to a complete withdrawal. Having an uncorrected error on the arxiv can damage the authors' reputations, so they should all respect the importance of removing the mistake. They should all work together to make the repair so that a new correct version can be posted as soon as possible.

At times, a new coauthor may be needed to help resolve an error. Sometimes the new collaborator can be added as a coauthor to the whole paper, or they might write an appendix or they might write a separate paper that is then cited to circumvent the error. All coauthors must be involved in this decision.

Note that if a paper has an error found after publication one usually submits an erratum to the journal describing the precise mistake and either providing a corrected statement or proof or withdrawing the paper completely if it unrecoverable. A similar comment mentioning the errata can be added to the arxiv posting. Or one may repost to the arxiv a new copy of the prior version with the errata appended at the end.

As a final note: never withdraw a paper from the arxiv unless it is incorrect.  Even if you have completely revised an article and changed the title and written stronger theorems (or plan to do so), do not withdraw the paper.   Just replace it with the new version.  If you've divided a paper into two papers (as journals sometimes request one to do), then post the first half as a new version, mention in the comments that the second half is now elsewhere and post that second half as a new article.   If possible keep the same title and write Part I and Part II so both halves are easy to find.