Battle of the Gauntlet

You are forced to bring Yuun and Tanno Vik for the actual fighting aboard The Gauntlet.

You have a choice regarding who to send to the bridge (mission being 'background performed' by two team mates while you handle things with Yuun and Vik).
If you send Elara, she will be hospitalised until your next class mission "Operation Talon" is completed (after completing Talon, that's when I got her back at least).
If you do not send Elara, I understand from other comments that Jorgan will get the "Hospital treat" in her place.

I chose to gear Yuun with my ALT earpiece, implants and headpiece for this, and if you need to buff Tanno Vik's gear a bit, be sure to switch items as soon as you can see him (during the event (I'm not actually sure you can do that mid event as I haven't tried, but if you can and all)), because in the cutscene where you meet up with Vik, Yuun becomes unavailable, and even if you cleared the whole thing with Yuun, there's still a golden mob spawned for your return path.
(I managed the return trip with Vik in his "birth gear", so it's far from impossible, but I'm pretty sure it'd been helpfull).

Make sure not to "park" any "vital" alt gear on Dorne or Jorgan, (whoever will be indisposed), as their gear will be out of reach untill "Operation Talon" is completed.

I'll just respond to my own question.

Apparently some real life time has to pass, as when I logged in this morning I had a new quest called "Casualty of War", that simply required me to speak with Aric aboard my ship to get him back into operational status.

The person that is down for the count due to fighting on the bridge - When does he/she recover?

In my case I left Aric there while ordering him and Forex to pull back, but he got hit hard and is currently unavailable. I completed the R&R, and have done The Talon as well, but he remains unavailable.

Which is a shame, since I am an armsmech, and I used him mostly for crafting.

First note: Whenever you cross the bridges over the Gauntlet's huge loading bay barrel, double check that there's not a blast coming. If you die on the bridge, you resurrect down in the shaft and you're stuck there. Happened to me. Had to use Emergency Fleet Pass to get out and travel back. Thankfully the instance had not reseted and mobs hadn't spawned.

Second note (spoiler alert!): One of your teammates who you'll send to fight on the bridge will apparently be injured in any case. Mine was Elara Dorne. I sent her with Aric Jorgan. Elara recovered quite quickly however after completing the next questline and came back with full strength.

Letter from Jonas Balker after completing this mission:

Leads to Charge to War.

I left the fight to get gear for Yuun. I had gotten him and outfitted him with custom gear that was empty ... going to fill the items with "hand me down" modifications. I never fight with any companion except Elara. We are invincable. Died several times until I got accustomed to trying to heal. I don't have enough talents for healing and had to use cc and try to kill before dying.

When I came back, the mobs had respawned but the first control objective was still triggered and unreactive. Finished the other control objectives with Yuun, but after an hour of trying to get to Tano Vik, I left the instance and reset the quest. Back to Coruscant to do it all over. This time, the door needed to get to Vik was operable and not locked. As with every other tedious mission that makes you run through the Coruscant Space Port ... you must. No going the fast way to Fleet. You must board at Coruscant Space Port and travel to The Gauntlet.

This would have been an easy time if I had Elara, but the mission only allows you to have the appropriate squad member as assigned. I now know why I use no companion but Elara. And she never wears pants ... so pretty.