You can install our site as a web app on your iOS device by utilizing the Add to Home Screen feature in Safari. Please see this thread for more details on this.
Note: This feature may not be available in some browsers.
Unless model S is completely different than model 3 or Y, you should expect the airpods to stop charging as soon as the car goes to sleep, unless you do something specific to keep the car awake.
The apple tech specs I find on-line for AirPods 2 suggest the case has a 1.52 wat-hr battery with each AirPod having about 0.93 watt-hr each. So if all three were totally drained, you'd be looking at a combined total of less than 2 Wh (watt-hr).
To put that in perspective, driving one mile in your MS 100D probably is about 300 Wh energy use. So that says you'll use the equivalent of about 35 feet of driving range to fully charge your AirPods.