You can plant them out, but water them well. I found it best to mark each one in the bed with a cane, because then, if you need to weed during winter, you can be careful where the canes are placed. They really do disappear in winter, and young ones can be quite wispy when they first come up in spring.
I tend to mulch with well rotted muck or kitchen compost, or even cheapo bagged stuff. Not sure about the wood chippings though? wouldn't they maybe rob the soil of nitrogen? Asparagus needs a lot of nitrogen to grow well.