Echoing a sibling comment, lots of landlords require it now, and the basic packages that insurers offer you as a bundle with auto or other forms of insurance are pretty decent, depending on state.

Typically seems like $100-200 per year for coverage that would handle the loss of most of one's possessions, provided you don't get screwed by "well, you don't have the receipt" or "we only cover water ingress, not floods or leaks".

Probably a lot? I've moved around a bunch over the past 20 years, so have had several landlords. I think all of them for the past decade have required proof of insurance when signing the lease. I don't think anyone I rented from required it before 2018ish