So, the utility belt says it costs 7 pts, plus 1 point for each slot you activate plus 1 point if you activate the costume slot. Now, each of the items are also 1 point, so does that mean that if I run the belt with 3 slots active, the costume slot active, and use 3 items and a costume for those slots, the cost is actually 15 points? Or does paying for the active slots also pay for the point cost of the items/costumes you take, making the cost of that setup just 11 pts?

The items you add on don't actually cost points. You pay 7 for the belt, 1-6 for the slots, and 0-1 for the costume. The max points it can be is 14.

Also, you don't have to attach the objects/costume until you begin the game, so you can use different pouches in the slots from game to game.

Exactly what I was going to say. You can pay the cost for 5 slots, have only 3 items, and add an item to that slot when the "parachute" number comes up. It's one way to get more than 3 of the same item on the belt (although it's playing a completely random game in that aspect, but for a point or two, it doesn't make much of a difference).
Also, if you don't have a costume in it, you can use it on a LOSH figure and still be eligible to use Legion Lost (and the costume can always be added later if need be).