1. An obvious way to select the weights is to have them be powers of 2, from 1 to $2^{99}$. Every positive integer has a unique binary representation, so no two different sets of weights can be have the same total weight.

2. There are $2^{100}$ possible sets of weights. If any two of them have the same weight, then they can be balanced (removing any weights that show up on both sides), so all of the sets must have different weights. To have the smallest possible total weight, the sets must weigh $0,1,2,...,2^{100}-1$. The binary construction has a total weight of $2^{100}-1$, so this is the minimum.