Does "Added Cold Damage Support" add the cold tag to the supported gem?
I don't think so.
To my knowledge you cannot alter gem tags. Your setup will add cold damages to every hits of the main skill, but will not change the attack skill to a cold skill.
In case you don't know, there are very detailed informations about every equipped skills in the offense section of your character screen. There you can take a look to the % crit multiplier of tornado shot an see if it is affected directly.
(If you can test it, please tell us the results!)