it forgets the ones used a lot, and retains some of the oddball ones

If you use them “a lot”, wouldn’t they be in the 15 ?

Is there anyway to delete items of of the search history window?

Not on-demand, no.

Or a way to put often used phrases in there like favorites list, etc.

This is kind of a frequently asked for feature.
Sadly, no one has come up with a great idea for implementing it, or perhaps no developer has tackled it.
I think a workaround is to keep a file tab dedicated to such “phrases”.

Notepad ++ and Admin mode keep different search histories, it would be nice if they were synced (same).

That seems like it would be a security breach.