Then I could keep his/her deposit money and any capital improvements put into the building.

We charge a deposit on direct bookings to offset this risk. 2.

I'd have to double the security deposit, raise the rent by 20%, and double the cleaning fee to even make it worth considering-- which I would if people were willing to pay but the market doesn't support those prices.

Here in NYC you will get laughed at if you ask for a T12 on deals that go into hard deposit immediately. 

5) Is one month rent security deposit enough nowadays?

This is very odd especially if there was no deposit and work was performed .

My tenant, who received the company check, addressed to our company, was able to cash/deposit it at their big name bank.

@Andrew Steffens has a deal on Helix mattresses you might be able to take advantage of.The list is almost endless.

When we moved out, the landlord insisted on keeping the entire security deposit of $1,000. 

However, given our current state I'm preparing to set-up my own website and/or post on various channels that don't appear to offer the host insurance and I know asking the Guest for security deposit or to run background checks can be a deterrent for short stays.