All great questions, would love to know that too. Don't think any of that information is available yet though. Didn't read anything specific from hands-on reviews either, other than info that's already on Apple's page.
I want to know more about these. If I pop open the case next to my watch, will it connect there? My iPad or my iPhone? Will it go for the watch everytime since I am wearing the watch?
In the Keynote they said it switches to the device you start playing audio on. Which could be problematic if you have them paired to both your phone and Mac, but you don't want your Mac to use the AirPods if you just want to use regular speakers. But I'm pretty sure they have thought about those scenarios.
Will battery for the AirPods and the case show up in the Battery widget so I will know when to recharge them or the case?
No source on this, but I can't imagine they won't. As far as I know any BT device that's paired to your phone shows up there with their battery status.
Will they use my AppleID? So if someone steals them or I loose them I can live happily knowing someone can't use them?
That would be great, but tricky. They can't authenticate every time you want to use them. There doesn't necessarily need to be an internet connection on the device you're pairing them to using BT. But maybe if you mark them as lost (in iCloud or wherever), the first time they connect to the internet again you need to unlock them on one of your known devices. But it's only a partial solution.
About stealing: I don't think it's a bigger problem than most other BT earphones/headphones though. If you have the ones that are connected with a single wire that runs around your neck, it's even easier to just grab the wire, yank them out of someones ears, and run off. High-end Beats? Easy to grab, go and resell for a good price.
If they fall out when you're using them, you'll know. If you lose them in your house or wherever, just take better care of your stuff. No different than other small items, like your keys. But I bet most people will keep them in the charger anyway, so they're together and in a nice box that's less likely to get lost.