I checked the wiki and found conflicting opinions. So now I'm here to ask!
Feel free to flame if this is in the wrong section :flamethrower:

Does raising the block skill apply any benefit for people who fight with the hand-to-hand skill? The only formulas I could find involved blocking with a "weapon" (I'm guessing meaning with a sword or bow) and with a shield.

Yes it does. You′ll get your disarming and knock-back features with hand to hand as well

The construction set wiki is probably most reliable on this since it examines the actual behind the since variables.

It appears that blocking with your hands would be one quarter as much as if you blocked with a shield. That takes into account your blocking skill:

http://cs.elderscrolls.com/constwiki/index.php/FBlockAmountHandToHandMult