Eagle eye, steady hand, critical shot definitely do affect crossbows..

But I'm not so sure about power shot, ranger, quick show and bullseye.

Just doesn't seem worth getting any other perks than overdraw, eagle eye and steady hand if you use crossbows.

They all work.

All of them do, according to bethesda.

Do you know this for sure?

Yes. They all work. My damage goes up when i pick the damage perks. I recover more bolts when i use that perk, they all work fine.

The arrow recovery perk is under-rated when bolts are rare to find and expensive to buy. You can even get the exploding bolts back.