In contrast to what you might read elsewhere, you don't have to get your characters ready in any way. Just run into the cave, hit the 'enter' key, cast a light spell if you can, and orient yourself on the stuff on the floor. There's a bunch of stuff, but, for example, all I grabbed was the 'contestants shield' and a 'longbow,' and I immediately ran out again.
Return to Lord Markham, and claim your 1000 gold.
You need to find the following: a Seashell, a Wealthy Hat, a Longbow, a Red Potion, a Floor Tile, and an Instrument.
The Seashell can be found in roofless shack (x -67, y 16958)in the northwest, guarded by a horde of Dragonflies and Fireflies. I killed them by luring away a few at a time until I was level 2 experience, then blasting the rest with a firewand given to me by a guy named Mr. Malwick, who's around the Alchemist Tower (x 6343, y7976) You may have to rest once or twice and remember to check your stats for training and/or healing.
Alternatively, you can pay Sally 100 gold for a seashell. She's always found outside the Dragon Cave, around (x 14718, y 16641).
The Red potion can be found, also, in one of 2 ways. The more expensive way is go buy it from the Alchemist tower (x6351, y7755). Alternatively, if you have a character with the alchemical ability, you can mix your own red potion from an empty bottle and any red reagent, such as Boysenberries.
The longbow can be found in the Dragon's Cave, just as you walk right in, so you can grab it and flee. (x14861, y15255) There's a crude bow to be found in a crate, but that's not the bow the Judge is looking for.
The Instrument can be bought from a peasant for 500 gold. For example, you can go to Ailyssa, who's usually around (x12173, y4600), and she will sell you one. I think there might be another, but I know Ailyssa, at least, is always there.
The Floor Tile can be found in the Temple of the Moon, after you slog through all the bats, mice, and spiders, in the Altar Room proper. There's a panel to touch on the left side wall behind the altar, leaving the altar to the left, and you end up going into a library of sorts. (Also be sure and grab all the books and scrolls you can, one per stack of books.) One of the walls will have a slightly radioactive red glow... the tile is in a hidden book stack in there, along with more magic books and scrolls.
The Wealthy Hat can be found in the Temple of the Moon, in the Altar room proper, on the right wall. This time, go around the altar to the right, and you'll end up in a guard's bunkroom. I couldn't find anything on the beds, but the Wealthy Hat is in one of the closets. Be careful NOT to talk to the guards on your way out, or one will attack you for taking the hat. (Unless, of course, you need the combat experience, but he's very tough if you're low level, so I advise against it.)
After you have all the objects, return to the Judge and Lord Markham and hand them in to get your deed to Harmondale.