If your pet's health gets too low or its attacks aren't very effective against a certain combatant, you can always switch out your current pet with another companion on your team. Only pets that are considered active (i.e., those in your three Battle Pet Slots), may be switched in and out during the battle. Once the match is over, you can always change up your active pet team, which is useful if one of your pets is low on health or dead.
In Classic, playing different classes actually feels different in more ways than just their abilities, at least if you think about it a little. Playing a priest, you really want to find a group because that's what the class was designed for, which is both good and bad - you're stuck very slowly leveling if you're solo, emphasizing the frustration of the slow pulling, but having different classes require different gameplay can be good. If you want to play a priest you're going to have to be more social by default, just whisper the guy you see killing stuff next to you and go level together, you'll immediately have a better time of it (this part is particularly exacerbated in the beta, as there are fewer players).
Warrior should be a go-to choice for everyone who wants to get into the end game. Protection Warriors in Classic WoW were considered as one of the best tanks, Fury talent build was a great option for DPS while Arms were dominating PvP environment. However, Warriors’ strength relies on the character’s gear, which means that you might need to put some hours into finding the right items for your class. It is also worth mentioning, that Warrior might be one of the hardest classes to level up in World of Warcraft: Classic but rest assured, if you’ll manage to hit the level cap and find suitable gear, Warrior is one of the strongest classes in the game.
Within the context of Classic, the method most frequently proposed is to re-tune bosses to make them harder. Rather than copying the numbers, re-create the relative difficulty. Kind of like, when you first start working out, you use small weights. When you get stronger, you lift bigger weights. The WoW community is more knowledgeable, more experienced, has better hardware, etc. We're "stronger." So give us tougher bosses. There are a lot of forms that could take. Bosses were constantly re-tuned during vanilla anyway, so if Ragnaros does 10% more damage or something, is that really a problem? I don't think it is. What about resistance gear? back in the day, collecting resist gear was a thing people did. But the numbers and strategies are so well known today that only the tank really needs it anymore. Bosses could be re-tuned with more resistible elemental damage so that's important for entire raids to gear appropriately. That would be entirely keeping with the spirit of vanilla even if the actually numbers sitting in a database somewhere were different than they were ~13 years ago. If you want to go a little farther, a few people have proposed adding new abilities to old bosses, adjusting their timing, etc.