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1. The minimum STR for heavy weapons is 12, so I assume the drone would be able to use it, but would take a -2 penalty. I don't see anything in the drone section to indicate that it wouldn't. Bear in mind that it would also need the heavy weapons proficiency, so in total including the extra weapon mount, you'd need to use 4 of your mods to do it. Assuming you forsake any other mods, you'll be able to get full use out of it by level 8. You could easily upgrade the weapon as you get the proficiency mod for it, though.
2. There's no reason I can see why it can't, if that's what you want it to do. The hover drone automatically comes with the ability to fly all day every day, but you can have it land whenever you want. I'd rule that having a drone with a heavy weapon on your shoulder is going to affect you just as much as carrying the weapon yourself. If the drone automatically takes off in combat it won't affect you there, but might if you need to use athletics out of combat.