Start by pulling both levers to drop a rock, allowing you to cross the spike pit.
As you descend, take note of the blocked passageways. The first is to the right of the first ladder.
The second is to the left of the first rope.
The last is to the right of the water puddle.
When you reach the bottom, the Miner will throw out some items. You'll need the bucket and the shovel. First, pick up the bucket and head to the water puddle. Dip it in to get a bucket of water, and return to the Miner.
With a bucket of water, you can catch and douse the dynamite the Miner throws at you. Catch a stick, and head up to the first blocked passageway (the one to the right of the first ladder). Use a nearby flame with the dynamite to clear the way.
Upon entering, jump across the first gap to reach a Cave Painting.
Have a second character grab the shovel and join you in this passage. The one with the shovel should hop into the empty mine cart, and the other should push the cart out of the room.
Switch control to the shovel-holder. While riding, use the shovel to smack the track switch and guide the cart back to the miner.
Repeat the process of getting doused dynamite in the water bucket. This time, clear the way to the second passageway (the one to the left of the first rope).
Upon entering, the first thing to do is leap across the top platforms to a Cave Painting. Refer to the screen shot above for clearing the rest of the room.
Platforms A and B move with weight. The mine cart starts off on the ground level. Have a character stand on platform B to lower it. Use another character to push the mine cart onto platform B, then into the middle of the room. Have the first character hop onto platform A to lower it to the ground, and have the second character push the character onto it. Finally, have two party members stand on platform B to raise platform A to the exit. From here, you can push the cart out of the room.
Repeat the doused dynamite process one more time to open the final passageway (the one to the right of the water puddle).
Drop to the ground, and then climb the ladder to the right. There's a Cave Painting next to the levers up here.
Have a second character move and hold the mine cart under the claw. The first character can then hit the levers to move gold into the cart. The cart can now be pushed through the checkpoint on the left side of the room.
With all three carts returned to the Miner, hop aboard to crash through the bridge to the next level.