No, doors provide no cover. Once opened they remain opened even if closed.
I found this when playing a UFO mission and two soldiers either side of the UFO doorway. I opened the door to 'spawn' the enemies inside, which they did. Then I closed the door, hoping to limit their ability to hit me and force movement.
The next turn a Viper pulled one of my troops directly through the closed door with it's tongue attack, and ADVENT troops shot through the door at my other squad mates.
I don't think there's a way to consistently cause a car to explode sooner other than by doing more damage. Hitting a car twice with guns seems to work fairly well, or hitting with a powerful explosive, like a plasma grenade. Destructible objects have an invisible HP value and much like when damaging enemies there's a range of damage values it can take.
I believe in the ini files there's values for terrain HP and cover damage HP that you could tweak, or at least peruse to see what the thresholds are. I've seen a shot from a ballistic heavy cannon + ballistic assault rifle blow up a car (in a hilarious gif) so a 2 shot explosion should be at least as likely with mag or plasma weapons.
Best Answer
Hold Ctrl while selecting tiles to set a specific movement path.