Capturing Live Aliens muliple times?

After you have enough weapons for your soldiers there is no reason to capture them alive. Before that point however stunning aliens is a good way to get your hands on the expensive weaponry, such as heavy plasmas and plasma rifles.
Also capturing more than one of the same exact type seems impossible in one mission

It's a good idea to stun when possible, but if you have already researched them, I wouldn't suggest going out of your way to stun them.

Well actually late game requests sometimes ask for weapons so collecting some when you can is a good idea.

The only reward for capturing more is to get your hands on the weapons they are using as they remain intact after battle.
But you do need the research for said weapons in order to use them. Other than that there's nothing else to gain.

As many noted, Free Weapons are the best reason for repeat captures. My second game I'm capturing my quota first if I can before killing, unless the odds are bad. I just raided the alien base with a full 6 man squad armed with plasma pistols & light rifles, Rifles haven't shown up on mobs yet. 18 more plasma pistols & 8 light rifles in the bank still.

How does one revive a critically wounded soldier in battle? I can only seem to stabilize them.

Made list over what each alien race gives on capture/kill without adding any spoilers. But the interrogation is only a one-time occurrence as any further capture of that same race only leads toward giving you their weapons intact which is useful since you dont have to build them once they been researched. Killing them gives you weapons fragments instead of the weapons they carry. But once you have enough of one weapon type, killing them gives you more since you can use those weapons fragments for research and build things that living aliens don't give like alloy cannon for example.

Sectoids
Armament: Plasma Pistol
Sectoid Autopsy allows production of Uplink Targeting (Aim)
Interrogate Sectoid awards a research credit towards Beam Weapons

Sectoid Commander
Armament: Plasma Pistol & Alien Grenades
Interrogate: Research credit towards Psionic tech
Autopsy unlock Psionic Labs facility

Sectopod
Armament: None that can be salvaged
Autopsy: unlocks advanced Constructions project in the foundry
Interrogate: Can not be taken alive

Thin Man
Armament: Light plasma rifle
Autopsy: Improved medikit projekt available in the foundry
Interrogate: UFO technology research credit

Outsider
Armament: Light Plasma Rifle
Autopsy: Gives no body when killed
Interrogate: Story related research

Mutton/Muton Elite
Armament: Plasma Rifle/Heavy plasma
Autopsy: Opens up the Foundry project, Ammo Conservation/Research credit toward plasma technology
Interrogate: Research credit toward plasma technology/Elite gives no interogation

Floater/Heavy Floater
Armament - Light Plasma Rifle/Heavy Plasma
Autopsy - Can build Defense Matrix boost/Advanced repair project in foundry
Interrogate - Research credit toward basic Armor/Heavy Floaters gives no interogation

Drone
Armament: None that can be salvaged
Autopsy: Drone Capture project available in foundry
Interrogate: Can not be taken alive

Cyberdisc
Armament: None that can be salvaged
Autopsy: Can build UFO tracking boost
Interrogate: Can not be taken alive

Chrysalid
Armament: None that can be salvaged
Autopsy: Can build Chitin Plating
Interrogate: Can not be taken alive

Berserker (Muton)
Armament: None
Autopsy: Can build stims equipment
Interrogate: Research credit toward Advanced Armor

Ethereal
Armament: None
Interrogate: Research credit toward all technology
Autopsy: Can build Mind Shield