Set pieces aren't really more common than others, they're all in the "common" category(with 3 categories, common, uncommon and rare, although that's not actual terms for them, just explaining, rare being shit like furnace and what not). That said some slots for some classes have more legendaries, and since they all share a chance to drop, that makes some set pieces less common than others, kinda. Like say you have 2 boots and 10gloves, all common, 1set piece and rest is legendaries, every time legendary boots drop, you have a 50% chance it's your set piece, but everytime legendary gloves drop, you'd only have a 10% chance to get the gloves. That's kinda where the less common thing happens, although it's generally a difference between 15% and 12% or so, so it's not like massive.
Legs in general are the slot you're most likely to find your set pieces from. That's cause there's not that many legendary legs.
Other than that, ancient is a flat chance on everything so it shouldn't affect it much, but RNG gonna RNG, you can play several seasons and never see a slot ancient. You might also not remember the ancients you got that had shitty stats. Unlike legs which are very easy to fix with the enchanter since you only need a few things and most of them roll sockets by default on set pieces, gloves need very specific stats and it's very easy to get ancient gloves that just suck because they don't have either crit or crit dmg so you can't enchant the 2nd one or the one that rolled is really low so even non ancient "perfect" gloves are a lot better.
On the T13 discussion, and the powercreep, it seems there's not a whole lot of buffs in next patch, so far. In fact, they might be nerfing shit, since they said they're planning to nerf support shit heavily and that it might affect solo skills too(buffing skills that is, like wolf for DH and shit). Group GR is probably going down next season if they do get their shit right/in time. That's why I think T12 is probably going to be a better pick, at least for solo rifting, at least until you're like REALLY geared. If grouping, T13 is probably going to be what you go with.