Exactly this. Don't look at the renewal proces, look at its output. It'll work for all certificate sources and catch other potential errors too (eg the webserver reporting success but not presenting the new certificate)
That'll work too. The idea was to put your own infra in place if you really need that, and it's not very hard to do it, even with completely self-hosted stack.
I imagine this is best left to third parties like the recommended service linked in the post. I assume that there's also a whole deluge of other services that have similar offerings.
at least for iOS there are no costs associated with using Apple Push Notification Service (APNS) but depending on the way you use it you either need to pay for the infrastructure that sends your notifications to Apple or for a service like OneSignal which does that for you. Not sure what the volume of LE is but I am pretty sure it's a smart move to focus on their core "business" (providing certificates) and let other handle expiration notifications.