Is UsernameRobot as username ok?
No. Such requests will generally be declined because the word is only allowed for bots in some wikis.
Can I ignore the Name too close than... warning?
You can ignore the warning in special cases. For example when a requested username contains some special characters the output of AntiSpoof might be wrong, see phabricator:T105626.
How to handle multiple rename requests
The policy only allows name changes if there is a good reason to do a rename.
We don't allow users to change their username multiple times.
Ask the user if they really wants to change the name again and notify them that this will (likely) be the last time we change the username.
Are zero edit-renames allowed?
They are allowed. Some users use Wikipedia accounts for saving articles to reading lists on Wikipedia apps and they hardly make any edits. Earlier such requests were rejected with a claim that they save server resources, but that's no longer true  and we shouldn't be worried about server resources.
Can inappropriate username be renamed?
The policy does not allow such renaming. Apart from that, it would be a lot of work to hide all rename logs globally. Such usernames should be globally hidden by a steward. You can file a request at SRG or email stewardswikimedia.org (especially for user name that should be suppressed).
Is there a limit of edits for accounts which are getting renamed?
No, there's no such limit. Earlier there used to be a limit of 100,000 edits after which we had to coordinate with sysadmins, but after technical improvements it is no longer necessary. We even renamed an account with 1,000,000+ edits  after these improvements.
The rename takes longer than usual?
Yes, It might be possible that a rename job runs for hours. If the job runs for more than 3 hours, it is suggested to file a report on Phabricator.
The upload logs on commons are not updated?
This is a cache issue and should be solved after some hours. You can purge them by hand if you like.
Is login during the rename progress possible?
During the rename progress it is not possible to login. Users who attempt to login will see a warning.
Can I rename myself?
Other common questions
Name too similar to existing accounts or used to username of someone who got renamed
In cases of similar usernames it's best to follow ACC's similar usernames flowchart.
In case of similar usernames, it's also useful to check if the username requested was username of someone else. This can be done by going to Global rename logs and entering the username in "Previous username:" field. If username is found to be name of someone else who got renamed then such request should be declined unless it's the same user.
Technically unacceptable accounts
To know technical restrictions on usernames, please see this guide. Also MediaWiki doesn't allow registration of number only or mixed script username but antispoof will not show warning in such cases, so it's best to check them by going to Sign up page.
Right to vanish
Please see Right to vanish for such cases. It's best to leave such requests for someone who's well aware of local policies or practices and can read the language since such requests are often used to hide or obfuscate bad conduct.
Some project allow such names (i.e. Wikimedia Commons) some even don't allow names like "John at XYZ company" (i.e Italian Wikipedia) so again it's best to leave them for local renamers.