The batteries in a typical UPS are SLA, which wear out much quicker if they are deeply discharged. So first benefit is longer lifespan of the batteries. They are wear items anyway, typically 3-5 years at best, so the nicer you can be to them, the better. Second benefit is how long before you can safely use your server again. Recharge rates are generally about 1 / 10 to 1 / 20 of the discharge time, so if you are running on batteries for 10 minutes, it's probably going to be 2 or 3 hours before they are full again. You really don't want to get in a corner where the power comes back on for a few minutes and you think the outage is over and turn on the server, only to have the power go out again and run completely out of battery.
Sometimes you might get away with it, but uncontrolled power outs are notorious for causing file system corruption. You must always stop the array properly before removing power. That's the reason to have a UPS, to avoid losing power with the server still running.
As it should be. The last thing you want is for everything to start back up without your permission and supervision, especially after a power loss. Who knows whether the power is going to be stable, they might have to cut it back off to complete repairs that they didn't get done or the grid might not be settled yet.
Why? That would mean if someone stole your server, your encryption passphrase would be conveniently entered for them, and all your files would be exposed. Encryption is a powerful tool, but it adds complexity that may not be worth the hassle. For example, an encrypted filesystem is MUCH harder to repair if it gets corrupted, so having full backups is that much more important.
If the UPS properly supports it, yes, providing the server is the only piece of equipment connected to the UPS. Many people choose to keep their internet modem, router, and switches on UPS as well, that way when the power goes out they may still have a few minutes of internet if the outage doesn't take out your ISP equipment as well.