Does using bare hands increase 1-hand or 2-hand skill level, and do those weapon skills apply to bare hands?

I was wondering if by attacking bare handed, you increase your 1-hand (or 2-hands) weapon skill mastery.

I played through the start of the game unarmed (Khajit), and never increased my one-handed or two-handed skills.

[...]do the Khajiits's +15 claw bonus mean that every one-hand weapon below damage 15 is useless?

If only comparing pure damage, yes.

Does, as example "Backstab" skill bonus apply just to any 1-hand weapons, or to barehand too?

The sneak perk Backstab applies to one-handed weapons, which bare-handed is not.

Can I, say, do a stealthy istant kill from behind when unarmed? Like breaking the neck maybe?

You can sneak attack while unarmed (and with a two-handed weapon) for extra damage. If you do enough damage to kill the opponent in one shot, you may be treated to a special animation. IIRC, there's an unarmed neck snap one.

In Skyrim, Unarmed combat is not tied to any skill. The damages don't get better with skill level, only some perks and equipment can give you more hand-to-hand damages.

You can have more details about improving your fist damages in this question.

Khajiits actually have a base damage of 10 (alongside argonians) whereas every other race only has a base damage of 4. Unlike argonians, the khajiit get a bonus 12 (not 15 like the game states) unarmed damage, giving them a base damage of 22. This makes unarmed combat for khajiit very useful early game as it has higher damage than many weapons.