Great idea. I always wanted to try the challenges while they were going on, but I never found the time.
I just tried the No queen challenge. I won on my second attempt. I'm gold league, so I had it a little easier than you. :P
My strategy was to use mass muta. The idea was that mutas cost only few larva. Also, since they cost mostly gas it wasn't all that important to saturate my bases properly, so I wouldn't miss the larva from injects too much.
And tried part 2, Carrier rush. Here I won on the first attempt.