I have searched for it and would like to know what would happen if a pokemon were to die. Would the trainer go on and get another pokemon? Or maybe they would stop being a trainer forever?
I suck at finding specific cannon sources, but here's is my general understanding.
You are asking one of two things. What happens if pokemon (in general) die instead of fainting? OR What would happen if a pokemon (just one) died instead of fainting?
If you are asking the first, that's a bit opinion based, but you would likely see defensive gear for pokemon to protect them more, higher regulations on battles to ensure fights don't go passed certain limits, and more medical items to help critically injured pokemon to prevent them from dying.
If you're asking the second, the trainer's response depends on the trainer. Some continue to fight. (I would assume most. If their goal is to become a master, it's a loss they learn to deal with. Not that they become emotionless, but they learn to remember and move on and fight in their honor etc.) Some would possibly stop because they don't want to lose more people.
This can be seen as something similar to when people command a division in any kind of combat. You either continue to fight when you've lost a man, or you discover that kind of stress is too much for you.
There's always theories like this one that go over whether Red kills his rival's Raticate. This implies that pokemon CAN die and have died, with your rival being an example of someone who continues on anyway. And with a city like Lavender Town and the cemetery and the natural effect of aging (since there are baby pokemon), you can reasonably assume most pokemon can die of old age and, therefore, other causes of bodily deterioration/harm.
Hope that helps out some. It's a bit long because I wasn't entirely sure what you were asking. Feel free to comment and I'll try to explain more or find better sources :)
Next thoughts are based on manga, so I'm not sure if all of this applies to anime.
First of all, it is not that strongly dependent on pokeballs. Pokeballs were did not exist forever, you know. And even when they've appeared, there was first pokemon captured by pokeball, which means, trainer managed to capture it without other captured pokemons.
You see, it may also depend on attitude of pokemon towards particular human. If pokemon likes someone, it may willingly allow to capture him, or at least obey commands of human he likes. Last one even applies to pokemons, who belong to other trainers. Also, trainers may lend their pokemon to other people. In manga, several main characters lent their pokemons to people without pokemon to help them caught one (Red lent his Pika to Yellow to capture Rattata, Ruby lent his Ralts to Wally).
So, there are numerous solutions to solve situations, at least in manga.
In terms of trainer mental state - well, that heavily depends on character mentality, and is mostly same to real world situations. Will you take new cat, if your cat dies?