Hi @Karen3489 Welcome to hosting.
First and foremost, I am not sure where you read that Airbnb does background checks, because they do not. The only verification that is done is through the ID, which I believe is still optional. As far as the ID being a valid source of verification, there is a gray area, because we as host don't see the ID, and at times I have had guest book with obscure names such that are obviously just a moniker or nickname. For example, I had a guest book as "DJ Powers." I had to pry his actual name out of him, despite it saying he had a "Verified ID."
I have hosted quite a few stays and made four claims through what is now known as Aircover. The best way to approach claims is to add photos if possible then provide verification of cost, whether it's just a generic quote online or an actual provider estimate. I will give you an example. I had a couple smoke in the room. The drapes and throw pillows all had to be professionally cleaned. Obviously, this is something you can't provide photo evidence for. What I did was I found a local dry cleaner and generated a quote on their site. I then took a screenshot and uploaded it to my claim. I have done this a couple where the damage was not something tangible per say. As far as before and after photos, I have never been asked to provide before photos showing the item was in good condition, but I do think it is wise to have photos of all areas of your accommodation, in the event a guest denies damage or the support team asks for further photos.
As far as theft goes, you really have to pick your battles. Smaller items being lost or taken are generally seen as part of the business. I wouldn't be chasing down a towel or umbrella from a guest, nor would I submit a claim for them. If you are extremely tight on your budget, consider supplying lower cost items that won't break your bank if they get damaged or disappear here and there. For example, you can purchase decent towels, blankets and sheets at discount stores, but if you buy them at a department store, they will be a higher cost and thus equate to more of a loss if something happens.
Overall, the best thing you can do is set requirements for booking, and/or turn off instant book. You can elect to require guest to have things such as an ID, or prior reviews. By turning off IB, it also makes guest submit a request, which allows you to ask questions and essentially vet them prior to approval. This will help you weed out problematic guest and make for better hosting experiences.