Can I choose which filters appear in the ticket filter panel?

Yes. In the "Columns & filters" window (the "Columns" button on the ticket list), every field has a "Filter" switch that shows or hides it in the filter panel; the panel follows the same order as your columns. Some fields (Id, Subject, Reply count, and Created) are not filterable and show a "Not filterable" note. If you hide a filter that currently has a selection applied, the selection is cleared so it cannot keep limiting your list invisibly.