Hey I've been playing this for the last couple days and it's been good. A few questions though: some weapons like a horned axe I have, have multiple types of attacks that seem to be the same thing (eg hack, cleave, cleanly slash), is that the case or are there differences I don't know about?
Yes, there are some: 'hack' attacks are better at piercing armor, 'cleave' attacks go in deeper (good against huge creatures), 'slash' attacks have the highest combined contact and penetration (very good against huge creatures) and 'cleanly slash' attacks are shallow, but almost never get stuck in the opponent. In case of the horned axe, though, the 'slash' works more like the 'cleanly slash' does.
Then, I'm not sure if this is normal but I ended up defeating a necromancer tower using my horned axe mainly, and have like ~80 kills on it, though I don't seem to have increased my greatarms skill (1/3 gain from competent) or axeman skill (at all)?
The weapon skill for horned axe is set correctly in the raws, as I've checked, so the reason is most likely an in-game one. For instance, the skill rises with each successful hit - not the kills, if i recall correctly, so, if you've dealt each zombie in with a tiny amount of strikes, that might be the culprit of the small gain here.
Also for the moment I've been using this as my general armor setup for my adventurers, I think Im doing everything correctly?
Yep, quite a legit uniform, at least, for when you cannot find plate garment of good metal. If you happen across a decent 'plate armor' or 'breastplate', they are generally lighter and more protective than a lamellar armor over two mail coats; even more so for arm and leg protection (gauntlets, greaves and such), because those parts are more vulnerable to blunt trauma.
cave spider silk padded coif/armor
1 chain coif, helm (heaven found full helm to see if it layers)
A full helm wont layer over both the padded coif and mail coif - if you happen to find a full helm, then choose one.
For all of this I just generally go with the highest quality material I can use which I assume is tempered/dwarven steel though I've read that it's useful to have copper as an under layer to defend against striking blows?
Dwarven steel is absolute best for armor, tempered steel and (regular) steel closely follow. But yes, copper chainmail garments underneath the steel ones can strengthen this uniform against the heavier blunt strikes somewhat.