if you have one item left if you get knocked out or driven insane can you lose your last item

No.

When you are knocked out or driven insane then the number of items you lose is half your current total, rounded down.

So if you have 7 items, you lose 3 of them. If you have 13 items you lose 6 of them. If you have 1 item, you lose 0.

However ... your question asked "CAN" (emphasis mine), implying that you want to. As fat as I'm aware, the answer is still no. You can't voluntarily lose an item - you either have to find an encounter that takes it from you or possibly use it if it's a "use and discard" item.

Stenun is correct on all counts. You can't voluntarily lose an item unless something specifically gives you the option. And it at only one item when knocked out, you don't lose it either because it rounds down to zero.

on the same topic; what does and what doesn't count as an item for the purposes of discarding when unconscious or insane?

From AH rulebook, pag 16:

Important: When adding up items before discarding
them, count all Common Items, Unique Items, and
Spells. In addition, the Deputy’s Revolver and the Patrol
Wagon both count as items. Allies, Skills, and other
cards do not count towards this total.