Scenario IV - Glory, or not?

Tricky, actually. I'd probably back down. I wouldn't be the kind of trainer to push my pokémon too far - we could just challenge the gym again at a later point.

However, I also wouldn't like my pokémon to learn that they could chillax in important battles and not give their fullest, because when things get tough, I'll let them off the hook. So it's not an easy choice o:

Depends a lot on what this Pokemon is, how long we've been together, who we're up against, how the rest of the battle went, etc.

The topic doesn't mention whether items play a factor into this or not... so my most natural answer would be... I give my pokemon a potion and we continue.

If items are not supposed to be in this situation, then we will continue anyways. Losing is an important part of life, but giving up is a sign of weakness. I would rather my pokemon lose while trying their hardest instead of giving up when there was a possibility of success. Who knows, maybe the Gym Leader's pokemon has a weakness or can be swept up into a play that it can't recover from, but you can't exploit that if you take the current state of the battle at face value.
I have had competitive battles where my entire team would be finished off except for my last pokemon and the battle looks one sided, only to come back around with my last pokemon and pull an unlikely victory from the clutches of defeat.

To give up in that fashion is to show weakness to your pokemon and lack of trust in your pokemon, and that is much more harmful then being defeated in one battle.

If its definitely a losing battle then I would take the chance to go away and train more so I can beat them. But in the games I also have won against the odds with my last pokemon.

Depends on the situation. Pushing your pokémon to their limits is really important and you should trust your pokémon, if they want to continue the fight. But the downside is that there's a small chance that your pokémon gets seriously injured and you shouldn't risk it. It depends on whether the trainer takes the risk or not. I would do it, because you should trust your pokémon.

At this point I feel like I'd know my Pokemon well enough to decide whether it could continue, which is something I wouldn't know well enough at say, the first gym. Honestly, if the gym leader was down to his/her last Pokemon too, I probably would let my Pokemon carry on fighting , cause there's always a chance, otherwise, I'd just back down.

If its a pokemon I'm relatively new with or dont know as well, I would forfeit. If its a trusted member of my team and I've spent a lot of time with it, I would let it fight if I knew it could persevere.

I'd forfeit for the time being, retrain, and come back to fight back harder.

In the games, I would have pushed it. That's how I won my very first gym battle against Brock!

But if it were a living, breathing Pokémon? I wouldn't have the heart to go through with it. I'd forfeit, train some more, and come back even stronger.

Which Gym Leader are we facing here? If it's Drayden or Clair I might consider honouring their request, recharging and battling another day.

If it's one of the easier last Leaders like Volkner I carry on and wonder to myself how I managed to have such a tough battle against a weak opponent.

I know that my Pokémon can take it so I guess only if it forces me then I would continue battling but would stop the battle if it got highly damaged.