First round.
Assuming there are 4 scum and the districts are totally random, if I'm correct there is a 14/15 * 12/14 *10/13 ~ 61.5% chance that they are in 4 separate districts, and a 3/15 * 2/14 * 1/13 ~ 0.2% chance they're in 2 pairs, leaving a 38.3% chance they're in one pair and two separately.
4 districts: 61.5%
3 districts: 38.3%
2 districts: 0.2%
With regards to mass-claiming, overall we have no way of knowing if it's helpful or not (although each player obviously has some piece of the puzzle). I believe closed setups are generally designed with the intent that claiming too early is bad for Town. I think it's usually better to give those with useful roles a couple of Nights to use them before we guide scum in killing them off.
About claiming to our partners... I don't know. I'll think more on it. About claiming our items, I don't think it's too hard to guess at what a bunch of them are, and which ones scum want to get rid of.