1a)
No. Drones cannot default on most things; if they need an autosoft for it, they cannot do it without the autosoft.
1b)
No, there's no "act correctly" roll. That kind of rolling would really slow down the game and be just make drones have a chance to say "DOES NOT COMPUTE" at every command, which would be lame. If you command them to do something, they do it. If it requires a specific Skill, they need an appropriate Autosoft. If it would just be an action with no roll, they can probably do it. If it's just attributes, they can probably do it. "When faced with something novel or unexpected, or a complicated command, a Pilot program must make a Device Rating x 2 Test against a threshold set by the gamemaster based on how confusing the situation is. If it fails this test, it blithely continues doing what it was doing before, or simply stops entirely and asks for instructions." But this is not for basic tasks like telling a drone to fire a gun, this is explicitly for edge-cases that don't quite make sense.
1c)
There's no test. A drone autopilot using a Targeting Autosoft simply rolls Pilot + Autosoft Rating [Accuracy]. Bottom of page 183.
1d)
Pilot x 2 if it's on Autopilot. If you used Control Device and are remotely operating the drone (but not jumped in), you use your REA + INT.
2)
When you're jumped in, you also use REA + INT for defense. There's no Limit for this roll, because there's no Skill rating involved, only Attributes.
3)
Just about everything, yes. When you're jumped in, Vehicle Actions are Matrix Actions. Firing a mounted weapon with the Gunnery skill is a vehicle action. Defense Tests are not actions. Resisting damage is not an action. Those don't get the hotsim bonus or the Control Rig bonus.