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

You can sort the Queued Notification search results as you see fit and then save that as a saved search and make it the default. That would avoid you having to change the GLPI source code.

https://glpi-user-documentation.readthedocs.io/fr/latest/first-steps/saved-searches.html?highlight=saved%20search

Completed for the next main release GLPI 10.1.0.

https://github.com/glpi-project/glpi/pull/13926