The whole point of a "challenge" is to make it harder on yourself, or to see how much grinding you can skip and win on pure skill alone.
Here are the things you will absolutely need:
- Autopsy (unlock arc thrower project)
- Arc thrower designed and built
- Alien Containment built
- Live capture of Outsider via ARC thrower
- "interrogation" of outsider to detect alien ground base.
- Destruction of ground base, capturing ethereal comm device
- Sectoid Commander autopsy (unlock psi potential)
- Build psi lab and start testing soldiers for potential
- Hypercomm research
- Shoot down the overseer UFO (it might be possible for an interceptor with anti-air missiles to stay in range long enough for a kill if you burn through every single boost you've managed to create from captured alien corpses; if not, you'll need to segue into Alien Nav Comp, Alien Power Source, and New Fighter Craft to build the Firestorm)
- Research the Ethereal device
- Build Gollop Chamber
- Have one psi-active soldier wearing psi-armor and have them use the chamber
- Complete the temple mission.
As to your additional questions:
You don't get a game-over until and unless eight countries drop out of the program. Of course, losing a country means losing potential income and continent bonus, and by powering through everything as quickly as possible you'll likely be critically short on, well, everything, most of the time.
You don't need to wait until you've got maxed out soldiers and the best possible gear, but while it is theoretically possible to beat the temple mission with just starting equipment, especially if you've done it before and know what's waiting for you at each juncture, expect it to be brutally hard.
You'll be up against two Sectopods simultaneously in one chamber, for instance.
EDIT Come to think of it, I'd absolutely recommend playing the Slingshot DLC if you have it at all. It nets you a high level soldier and a big chunk of resources very early into the game if you succeed, and potentially lets you bypass a number of dangerous interceptions.