For my Templar follower, does it improve his stats/damage to equip his weapon with an emerald, for increased critical strike damage? Or should I equip all his slots with rubies? Likewise for the Scoundrel and Enchantress, is there any reason to try gems other than the ones that boost their main stat?
Items equipped on your followers do benefit from socketed gems. The wiki entry on followers doesn't go into detail, but followers do benefit from equipment and gems just like your character does. You can test this by letting your follower attack enemies with and without emeralds, and counting the damage of criticals (which I have done, sadly).
Your play style, and that of your follower, depends on the gear you and your follower use, and vice versa. E.g., rubies will always increase strength (which is useless for say, a wizard), and emeralds will always give critical damage (which is great for everyone, unless you want to focus on something else). In my experience, critical strike chance gives you the highest damage increase, pound for pound, even more than critical damage. If your follower dies a lot though, maybe go for vitality, life on hit, etc (or opt for a legendary follower accessory with "Your follower cannot die.")
It should be noted however that for followers, the critical hit damage bonus is capped at +100% (beyond that natural bonus), so if you have a follower accessory with high CHD, a ruby in the weapon socket will result in higher damage.
Yes... but not very much
First of all, an emerald in a weapon does not increase anybody's critical hit chance, but merely the critical hit damage. This is the same for you and your follower.
For the other slots (shield/jewelry), you should stick the one that gives a bonus to their primary stat, as it is the only one that will increase their damage.
However, it's not strictly true to say that there is no benefit from using the other gems, as strength and dexterity still increases armor for an intelligence-based character, and intelligence increases all resist. But since the primary stat has both the offensive and defensive perks, you should go with that one.
Apart from the weapon (which is debatable), you shouldn't really be using socketed gear on your followers, since they benefit more from other stats (cooldown reduction, for example), unless this is all you have to give to them. Sockets in jewelry are only good if you can use legendary gems, which followers unfortunately cannot.
I know exactly what you're asking. And the answer is DO NOT USE EMERALD FOR FOLLOWERS WEAPON. Use a gem for their base stats or pure damage. Templar can use both weapon and shield slot for 2 gem sockets, so use both RUBIES for him.
For your Scoundrel and Enchantress use Ruby inside their weapons. This will increase pure damage stats.
Emeralds do not work very well for the followers because their critical chance is way too low compared to the Nephalem. So the Critical Damage Increment buff goes to waste for them. (But for the Scoundrel you can try Emerald becuase his abilities are build around Crits).
My advice is equip a socketed weapon on your follower and check their stats change with changing gems. Should be pretty clear.