In terms of Homelink, often I have had to hit the button several times to get the opener to operate. This has nothing to do with the Tesla - my other remotes have the same problems. Auto open/close doesn't actually know if the signal actuated the door... if your opener works right, it will do so. When my opener acts right (operates with a single attempt), the Tesla properly actuates the door operation. Except for the Homelink anomaly, mine works perfectly.
Here's what I did to make it right:
I started at the end of my driveway and setup the connection. This way, the whole opening process actually happens at the end of my driveway, where it should happen. By the time I drive up the 75' drive... the door is already opening. It uses the GPS coordinate of my driveway apron for the "auto-open" operation.
Regarding the "close" operation... that is automatically determined by the start point of the car, parked in your garage. You get in the car, hit the Homelink button to open the door, and put the car in reverse. Right then, the car should start the 20' countdown to auto-close. GPS position for this is irrelevant. It knows you a) started the car b) opened the garage door c) are backing out of the garage so it starts the countdown.
Hope this helps.