After searching the forum for this topic, it seems as if the only way to "clean up" my admin back-end is to prune the database of trashed items and categories. Is that correct? Is that something I can do, or should I ask my host support to do it? Just FYI, when it comes to technical matters, I am somewhere between level clown and level buffoon. (That is probably why I have so much stuff trashed.)

Mark Russell
Glorious Leader
EmptyPotato, LLC

Hail Glorious Leader,
If you are talking about 100 - 250 items you can manually delete them quite easily.
Simply select "Trashed" from the filters above and delete all the items in that list.
From the db, you should be EXTRA careful of what you delete since you might 'breaking' some aspects of your site or losing data.