Is there a way to auto-relist an auction after it ends as unsold?

Easy answer Rachel. No. Only listing tools like My Products and others let you auto relist unsold.

I don't usually take much notice of the suggested category when listing as it's often not best and I usually list by duplicating an existing product then modify, but I do search listings afterwards to make sure my listing appears in searches. I haven't checked if searching matches the suggested category when listing but it pays to do your own searches on site so your items can be found and aren't excluded when searching as the suggested category(s) snap takes hold.