1. If you bring it down some of them are killed in the crash. I don't know if just damage affects terror mission, although base attack missions see an enemy casualty based on damage. Note that crashed terror mission attacks tend to be more dangerous, however, as there are few locals to soak up damage/delay the enemy, and there doesn't tend to be as much cover as in a terrorized city.
2. Carriers can be shot down. However, there are certain combinations that are not intended to be possible to shoot down because they are escorted by extremely dangerous craft. "When" is subjective. I think it's possible to do it with starting craft (assuming there are no murder escorts) if you're prepared to have a lot of them shot down, but I have never tried that myself.
3. A single remaining alien is capable of slaughtering any locals, so yes, pulling out of a terror mission is a failure. Pulling out of a UFO attack mission is a failure as well. In both cases you can salvage marginall amounts of gear by placing it in soldier inventories. Note that any dropped gear (including inside the drop ship) may get lost, so don't free up inventory space by dropping valuable weapons on the drop ship floor!
4. I use heavy save scumming at the "easiest" difficulty and terror missions are a pain (I don't get real terror missions any more as I shoot the craft down, but the downed craft I'm currently slogging through is plain silly (and I'm still not sure I'll make it). I'm just waiting for the final piece of the puzzle to be made available by the RNG to enter the end game, so that's at the very end of the campaign. I haven't failed any terror missions (or downed terror craft), though (save scumming and an exploit!).
Note that every kinetic/energy hit wears down armor, so even if all the damage was resisted, the shot still removed a small amount of armor (15% of the nominal damage), so resisted hits still contribute towards the damage of following hits. Smoke is useful to block enemies from firing at you, forcing them to come closer before they fire. It may buy you time and can protect you from long range shots.
I made a test of a Division Mk-3 rifle vs a normal Andron, and it took over 30 hits to kill it (save scumming to make sure every shot hit using aiming to increase the accuracy from abysmal to lousy. Two soldiers were participated in firing at a camping Andron, and no other enemies were active outside of the (non terror) craft at the time, so I had plenty of time). I believe the first damage was cause around hit 22.
Thus, kinetic weapons contribute marginally against Androns, but it's better to aim them at softer targets (they're useful against small drones, for instance).