Whether or not you would need to maintain a list of users currently depends on where you are going to initially publish your app. If you publish it in a public store—Google Play, Apple App Store, Windows Phone Store—you won't need to maintain a user list in AppManager as it will only manage the update packages that you push to your app.
If you need to publish an app in AppManager, have users install it, and then start pushing updates via LiveSync, you would need to make users register in AppManager for the initial app installation.
Note that this is about to change in the near future as we are already planning on implementing app distribution via AppManager without user authentication (exposing public links to the apps). When we release it, you will not need to ask users to register in your AppManager.
As for your second question, AppManager LiveSync has different behavior than what you are used to in AppBuilder. A LiveSync-enabled app automatically checks for updates on start up and prompts users to install them. If they refuse, they are again prompted at the next start. So, the check happens automatically and neither you, nor your users need to do anything to trigger it.
You can learn more about how AppManager LiveSync works here:
Update Your Published App with AppManager LiveSync
Let me know shoud you have further questions.
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