Implementation started for upcoming GLPI 10.1 release:

https://github.com/glpi-project/glpi/pull/14738

You can use recurring tickets with some TODO tasks in the template. Starting in the upcoming GLPI 10.1 release, you can also link assets to the recurring tickets.

Added in GLPI 10.0

In progress:
https://github.com/glpi-project/glpi/pull/14747

Added GLPI 9.5.0

The search does support a limited set of REGEX when using the "contains" operator.

https://glpi-user-documentation.readthedocs.io/fr/latest/first-steps/search.html#perform-an-advanced-search

Implemented in GLPI 10.0

Implemented in GLPI 10.0

Private followups and tasks can be utilized for this purpose.

"Saved search would be even better if you could manually order them so that they can act like mail folders"
Saved searches are able to be drag and dropped to reorder them. There was a regression in GLPI 10 that removed that ability, but it was re-added for the next minor release (GLPI 10.0.8). You can find the patch here:
https://github.com/glpi-project/glpi/pull/14519