I want to know whether block skill effects how well you block with one-handed weapon (no shield) and which perks in block tree are useful for that purpose...

No idea blocking is real hard with 2 handed and 2 handers, so its also is for 1 handers.
The hell even for 1 hander and shield its rather weak.
Learn to defense bash a lot to increase skill and survivability.

As far as I can tell, yes. At least I know time slows down when I block with a 2h.

Yes the block skill affects unshielded blocking.

http://www.uesp.net/wiki/Skyrim:Block

And blocking enchants effect it as well, even though the description is something like "block x% more damage with shields." Also, total blocking boost from perks is like 40%, you can get that from a single piece of enchanted gear (out of 4 possible) with 100 level enchanting, making that ranked perk almost useless. Right hand side of the blocking perk tree works with 2 handed, so I'd guess it works for 1 handed as well.