1- why would someon staying in a room leave an envelope of money under the mattress
No safe in the room?
2 - I would then be pretty sure that the name on the envelope wouldn’t be the name of the person staying in the room
What a highly odd presumption. I probably half a half dozen envelopes at home containing cards with cash or vouchers in them - gifts awaiting my use of them. I will take them with me when I need them. I suppose the reason the name might not match the resident would be that they were taking it as a gift (say,, to a wedding) and forgot it. But the hotel could still call former residents of the room, and ask them to name the person on the envelope and the amount.
3 - I would be quite sure that unless this is some super luxury hotel where they go or of their way to reunite lost goods with owners that the money would be handed in and forgotten about immediately
I have stayed in a £40 shitty motorway travelodge who called me to say there was a silver earring in the room we'd stayed in five nights previously. They were clearly working their way down the list of former occupants. Also someone handed in our dropped bank card to reception of another hotel. They called us in our room because my husband's name matched the card.
In fact my mum loses her wallet with alarming frequency, and has almost always been reunited with it in full - even by non 'super luxury' businesses.
4- how does anyone think that police or a hotel can trace an owner of an envelope of cash with just a first name? A name most likely not on their booking?
Initial on their booking? By calling the previous inhabitants of the room?
I hope you don't go finding anything lost of mine since you seem so inclined to look for excuses not to do the right thing...