Okay, I feel like I'm going crazy, but the battle in the Slums with 4 Trolls and 2 Ogres is ridiculous. The troll hit points are insane, and they regen every round! Also, I get that trolls in AD&D will resurrect after death if not killed by fire, but you can't even kill them like that from what I can find out. I tested by killing them with Flasks of Oil and Burning Hands, and neither keeps them dead. Is this known, or is it random or something? I am scared to even keep playing if I run into any other trolls later on that I will get stuck on.

This is a notoriously difficult battle. It's not just you, it's extremely hard. It's by far the most difficult battle in the game in my opinion (given your party strength at the time you're most likely to encounter it). You are correct that burning doesn't work the way it is supposed to, it's a lapse in the game.

Protip: If one of your characters is standing on a space where the troll was when it died, it won't be able to come back to life.

There are a couple of strategies you can try:
1) Come back to this fight later when your characters are at least level 3 (This is what I personally always do).
2) Hire some NPC fighters in the adventurer's guild in Phlan, and equip them with bows and arrows.

You’re not supposed to beat this battle early on. Return to the area later when you’ve gained a level.

Though it’s actually not that hard if you “cheated” your stats to max.

To have a reasonable chance, do the fight when you can cast Stinking Cloud on the Trolls. Sleep will usually work on one Ogre. Hire two Fighters in the hall of training and use them in the front line.
To kill the Trolls you need to use fire after they have gone down, not while they are still standing.

You can have a character stand on the spot where they died. Then they can’t get up.

Don't attempt this battle until all your characters are level 2 (for multi-class)/3. It should be the last battle you do in the slums. You can also go and complete Sokal Keep before doing this for a better chance at winning.

Hire two heroes and place them in the party so that they are standing adjacent to the trolls. Have everyone in the back equip bows if they can. As soon as any trolls go down, get one of your party members to stand in the spot they fell so they won't revive (you can also see if another troll will come stand in their spot, so you can potentially stand on top of 2 or more troll bodies). The trolls will eventually start running away when you're winning. Don't chase after them, just use bows to attack the last remaining few.

If your two hired heroes are still alive at the end, continue the battle and cast sleep on them to make them helpless so you don't have to pay out their share of the treasure/xp and take their magic two-handed swords :)