Would it be possible to return the money automatically after some period of inactivity, without any action by a mediator?
Do you mean as a protection against a case where operator is dead? Yes, it is possible.
For example, user is given nLockTime'd transaction which locks in a year and sends money back to user, but operator has nLockTime'd transction which locks in 11 months and sends money back into escrow.
If operator is alive he will release his transaction and thus will prevent release of funds back to user.
However, it doesn't save user from a malicious operator.
The site could also be run on any kind of shady web server in any part of the world, without a big risk by the user.
I don't think so. Operator still controls payoffs. If private key is stolen, I'm sure you will lose some of your money.
I personally would trust an automatic return of money more than any mediator, who could also disappear etc.
Well, I don't know, it's very unlikely both will disappear at the same time.
In my opinion, the worst thing in this scheme in general is that the winners have to wait for the losers to release their money. Of course this could be mitigated with the site keeping some amount of buffer funds available. And the buffer would be there just to speed up the payments, and not be an integral part of the operations.
Yes.
A different scheme is to use peer-to-peer bets with operator controlling payoff. (So-called contracts.) In this case no wait is necessary. (Winner can claim funds as soon as operator announces who wins.) Also people can simply get money back if operator is dead.