You can indeed do this with Sleep as Android - I use it in combination with a Pebble Time to feed Sleep as Android accelerometer data (for sleep phases), and then the watch vibrates for an alarm - falling back to an audible alarm on the phone if I don't deactivate it in time.

The only downside is that the vibration motor in the Pebble Time is a little bit loud, but still much quieter than an audible alarm on the phone itself.

You most certainly can. I have been doing this with my Moto 360 2nd gen and it works great. Unfortunately it turns out I hate sleeping with something on my wrist, so I don't do it anymore. It also slowly turns on my smartlights when the alarm goes off.

"Sleep as Android"? Is that an odd translation?

I believe so. It's an old app (2010?) and the devs are foreign I think, but the name stuck.

Maybe a play on "Do Androids Dream of Electric Sheep?"?

When do you charge it if not while you sleep?

I charge my watch when I get home since I hardly really need it between then and bed time where I can either glance at one of the clocks around the house, or take care of notifications and what-not on the phone itself. I then put it back on charge between getting up and leaving the house in the morning.

It sounds like a lot, but it's very quickly become routine.

I charge my fitbit when i shower