How does armor work in Morrowind?

You can mix armor types. The only drawback is, that you level them more slowly, if you wear part light armor and part no armor, then both will increase at half the rate, if you were fully unarmored or fully lightly armored.

Unarmored is generally considered very bad, as its armor value is very low unless you get it to enormous skill levels (which requires several powerful enchantments) and you deny yourself several pieces of armor, which could be enchanted. There's also a bug with unarmored (which is fixed by the Morrowind code patch mod), which requires you to wear at least one piece of armor to get any benefit from unarmored.

If you want to play optimally, then yes, you should focus on one armor type. However, as you picked the Nightblade class, which has both skills as minor, you may want to mix and match and level both, as both skills are important for levelling up. Generally for optimal play, you create a class, which only has one armor skill in their major skills (or in fact none at all as armor skills increase quite slowly)

Clothing is always useless weight as long as it is unenchanted. You can however enchant pretty much every piece of clothing to get a benefit. Armor and clothes can also be worn concurrently, only the glove and boots slot overlap, so no reason to not use clothes. If you want to mix and match light armor and unarmored, then those two slots would be the prime spots to go unarmored (also fits thematically quite well to keep fingers fully mobile and maintain a silent step).