Asparagus plants are ultimately large and with wide-ranging roots, so from mid July to September they can be bad neighbours to nearby plants. I usually crop early beets, peas, salads and beans near them in the spring then leave some bare space nearby. North south is a good orientation.
And with the weeds you describe, I would pot them on in the spring, to plant out next winter, even March 2015. Good luck with it and all your other growing, thanks for your comments.