I have had great gardening success with composting chicken poop I have come to treasure my chicken poop!
I have found that hay and straw take a long time to decompose so I prefer wood chips as bedding. Straw or hay won't hurt anything in your garden if you add compost with undecomposed hay/straw, however in my clay soil I don't like it because clay + straw gets very compacted (it is how they make mud huts / adobe in some parts of the world!) and drains poorly. I am not an expert, though so there could be some tricks out there to make this work better.
I don't have a vermicomposter now, but I did a long time ago. Now I just add the chicken poo + wood chip bedding to my compost bin along with veggie scraps, and I try to add as much green stuff (weeds, grass clippings) as I do brown stuff (wood chips, dry leaves), and I add occasional water, an occasional shovel full of dirt (for good microbes), stir it up every month or so, and a few months before planting time (now!) I add a package of composting enzyme stuff from the nursery to speed things up.
My compost is probably ~10-20% chicken bedding+poo and it is FULL of worms - the descendants of a previous vermicomposter. I don't know if adding straight poo to the worm bin would work - seems like it might be a bit intense for them, but if they have plenty of room to move away into other compostable stuff, maybe it would work. I would experiment and see how it goes.
I don't pay much attention to making sure the compost gets hot...I let it sit for at least 3 months before using it in the garden, and it seems completely decomposed. If this sounds wrong to anyone I'd like to know!
Happy composting!