Regarding the property "required on update" for custom fields. What is actually the applicabe use case for "update":
1) Updating an issue through 'bug_update_page' ?
2) Updating the issue status, eg: through 'bug_change_status_page' ?
Afaik, currently, the "required on update" is applied to both.

Is (1) actually an useful scenario?
Maybe it's cleaner to rename the option to "required on (any) status change", and enforce the required field whenever there is a status change
So, updating an issue, thruough bug_update_page, will check the required fields only if the status is changed.
This could be supported by the facts:

• updating an issue without modifying its status, is not a very significant update in the issue's life cycle.
• the "required on xxx" properties in custom fields are clearly workflow oriented

Another possible inconsistencies:
3) "require on update" does noes apply when the status change causes a "resolve" or "close" status transition, those have their dedicated "required on resolve/close" options instead.
I would say that "required on resolve/close" are subsets of "required on update" which should be enforced regardless of those options.
4) Assuming that "closed" and "resolved" are defined by status thresholds, one will be a subset of the other. As default: CLOSED status is considered both "closed" and "resolved". So in that situation, "require on resolve" should apply also when a status is changed directly to a closed status.
Even current logic seems to be wrong in code, as it's not checking if there are several status above "closed threshold":
https://github.com/mantisbt/mantisbt/blob/release-2.6.0/bug_change_status_page.php#L268