Do you lose the SD if you take a hit but then cancel it with green dice, tokens, etc., or only lose SD if you actually take a damage card?

Damage card / lost shield.

Only from a direct attack though. Area affect damage from a bomb or assault missile doesn't count.

Only if you are defending against an attack and take a hit, in the FAQ under stealth device.

Again, "hit" has a specific definition in the rules. Page 12 under the compare results section explains what it means to be hit. An explosion symbol is not a "hit".