This is actually a very old thread, FYI.
xgambit: 3) Is there any limit to how many you can have at a time? I know the next dude to answer said only one Lord quest, but...
That's the only limit: one Lord quest at a time. No limit on other quests. To be honest there aren't many formal "quests" in the game, most of it is just about exploring and figuring things out. The other quests that exist are tracked based on whether you have specific items or not. For example, if someone asks you to deliver an item to someone, then when you go talk to them, the game checks whether you have the item (I think your first party member has to be carrying it) and the quest will proceed. This is also why quests are repeatable. When you've finished it, you can go back to the original quest giver and they'll give you the item to deliver again (because you don't have any quest items in your inventory anymore). I think this design is due to the game's age; it couldn't actually track progress easily so they used inventory items to do it.
xgambit: Also, do I have to grab the treasure from a random battle before leaving the square, or can I come back?
You have to grab it before leaving the square! Otherwise it will disappear. But, you can rest first. As long as you don't move. This is very important because battles are where you will find nearly all of your loot.
xgambit: And what/where is the best way to get lots of random battles? Have been running around all over and can't find some, frustrated emoji here.
I'll defer to others who remember specific locations better. But I will say that I recommend simply exploring new places as often as you can. There's not much need to stop and grind for levels, except at the VERY beginning. Once you have some decent equipment and a level or two, just start exploring. If a place is too hard, go somewhere else. Exploration is a huge part of MM1 and it's really fun. And there's always somewhere else to go!
Pro tip: If you're just starting out with a party you made, your characters will start with nothing but clubs, which are terrible. But you can take better equipment off of the pre-made party. Sign in at the inn, take half of the pre-made characters back out with you and half of your party, transfer items from pre-made party to your party, then repeat with the other characters. That helps a lot in the beginning.