I have installed MCP that fixes Unarmored related bug, but now I have one question:

To increase my unarmored skill, must I be completely unarmored or can I have e.g. fur boots + common clothes and thus increase both unarmored and light armor?

I play vanilla GotY, but if you wear some armor, you will gain some experience in Unarmored, and some in whatever armor style you are wearing.

Exactly. Depends on where you get hit. The armor skill in that slot increases a bit. The hitpoints are random as far as I know. Or they might follow e.g. the same percentage that each armor slot covering your overall armor rating follow. IIRC the cuirass slot is 50%, so you shouldn't use at least any cuirass if you want to get good in Unarmored. Couldn't find the table right off the bat from UESPWiki at this time (if it's even related to this - again, might be completely random where you take each hit).

Thank you, I guess I'll play without cuirass, perhaps some fur helm or boots only.
I hope Unarmored was worth choosing as a major skill

Details about body part percentages coming up but first, about choosing unarmored - I like it because you don't have to repair armor if you don't wear it but even more so because the AR increases exponentially as it rises. This means once you get 100 unarmored you can make/buy enchanted items that add to unarmored skill to take it above 100 and your AR will skyrocket. I don't have the exact numbers at hand but theoretically if you could get say around 140 unarmored your AR should be comparable at least to glass, maybe more. I'll crunch the numbers sometime.

Body part percents: body = 30%, head, l arm, r arm, legs, feet = 10% each, shield = 10%, hands = 10% (5% each)

Example: You are wearing nothing on chest and glass everywhere else and using a daedric tower shield. On any given hit you have a 30% chance to increase unarmored, 60% to + light armor, and 10% to + heavy armor. Snobby comment: if you use medium armor, don't, it's lame. hehe just kidding use whatever floats your boat.

Thanks for detailed explanation, I'm using only light armor helmet now.

40% for cuirass, 10% helm, shield, greaves and boots. 5% for each gauntlet/bracer and pauldron.

One thing I also noticed while levelling up unarmored is that there is a different sound that plays when an unarmored part of your body gets hit... there's the slight "tink" of something hitting armor(Checking stats after hit showed armor skill went up), and the meatier "thwack" on an unarmored part(Checking stats after hit showed unarmored skill went up)...

I've posted the math before, but am too lazy ATM to dig it up again... basically, unarmored can be just as powerful as full Daedric, if you fortify above 100. Normal armor skills increase armor rating linearly with skill level, but unarmored increases exponentially. So, max out unarmored, get some CE Fortify Unarmored clothing, and you'll be just as well-defended as our favorite Mr. Fyr. AND - you won't ever have to repair anything.