Why would you need a special cleaning company for covid-19? I thought they found the virus can only live on surfaces for a few days to a week at most. So if the house is empty for a week nothing should be needed related to COVID-19 ( https://www.sciencedaily.com/releases/2020/03/200320192755.htm )
1 - Regarding filing an insurance claim, if there is damage to be repaired such as replacing all carpets I would yes.
2 - I would think the downstairs tenant already knows if the neighbors know they died. You don't have enough information yet and all you could do is cause more stress by talking about "what ifs" now.
3 - I think that depends on your state if the lease is passed on, but I haven't heard of that before. I thought it would go to an estate probate eventually for debts but I think that death is considered something that lets them out of any leases. They can pay you anyway if they want to but I don't think you can force them to legally.
I'd wait until you have more information if you have to replace carpets and if they died of coronavirus or not before doing anything, give it a few days or a week. I suspect you'll just be replacing carpets in that room if they had sat there for awhile and the rest of the house is probably ok. Don't go into the unit until you get the all-clear though!