2. Can a character wear a lantern? I assumed no after I read a forum thread (not related to this game though) about how impractical that would be. Best case scenario, the oil would spill out and/or the lantern would limit your movement. Worst case scenario, you set yourself on fire.
Well, if you can creatively come up with a way to do it, it's up to the DM. I see photographers wearing their flashes (including large diffusers) at conventions by attaching a boom to their backpack or harness, so I don't know why a lantern would be that different. Though if it were me DM'ing, I'd say, "sure, but you get a movement penalty," to keep it balanced.
4. How do you design an adventure? I have no idea what I can throw at my little party and what I can't.
This is a huge question that cannot be answered shortly. First, a general tip: create situations, not stories. Putting players (or rather, their characters) on a rail tends not to be very fun. Nothing in your adventure should depend critically on anybody doing anything in particular. There's ways to deal with this that are beyond the scope of a forum post, but there's a lot of good advice out there how to create adventures.
How much can you throw at a party? Figure out the relative probabilities of monsters hitting players and vice versa. Figure out each side's total HP. Figure out each side's average damage. That'll tell you how long it takes one side to whittle the other down. It's not exact, of course, and players will often do better by creative tactics. But if your party wipes out 10 enemies in 2 rounds, maybe the challenge is too low. If it's likely somebody dies in the first round, probably too tough.
In general, keep in mind that it's fair to protect your players' characters from bad luck. Three natural 1s in a row can take out a player or whole party, but I think the fewest players would object if you gave them some creative benefit so they have a small chance at survival. Maybe the PC dies but they can take over an NPC. At the same time, you don't have to (and probably shouldn't) protect players from stupidity.
Did I see you plan on playing solo? You can find random map generators online and just try it that way. There's ways to stock monsters and treasure, too, using random input. I think this would get repetitive after a while; it's how I played my first games, too, pretty much. But it's a good way to get to know the mechanics. Of course, you'll miss out on any real surprises and roleplaying, but that's of different amounts of importance to different people.