Hi,
Not sure if the answer above really helps you. I can't advise you how to check on Live status - as above the current live version can be viewed but you can't guarantee that this means that the platform itself is Live.
You should use the Microsoft365 admin centre status updates and email alerts based on these to see if there are any issues with the platform:

In terms of the application itself having errors - on newer versions of Apps (in the last few days) I have noticed this App function above the first loading screen:

From here you could put a function to notify you of errors.
I personally would advise against using email alerts for errors as reply above states, as your mailbox may become overwhelmed at some point, and instead would have these errors stored in a database, whether SharePoint or SQL, with columns for details of the error, time of occurrence, username of submitter, details of current variables within app. You could then pass this data through to PowerBI to be able to see at a glance if any new serious errors are occuring.
I would also suggest using this guide that @Anonymous wrote to keep track of how many Apps are in your system and their details:
https://www.linkedin.com/pulse/powerapps-using-flow-admin-connectors-manage-your-manuela-pichler/
I'd done a similar thing to what they've done on the 'get new powerapps' function except I don't email it to myself and I've applied it to 'all time' rather than the last 7 days then pushed it up to a SharePoint list, to keep track of which apps have bypassed consent, which use advanced device functions like webcam or location sensors, and am now able to then sort/filter by environment etc:

Hope this helps you in some way,
@iAm_ManCat