I'd agree with Lonelymum - it's probably coincidental. Sounds like one of the dread "stages" that would have happened anyway. I would just persevere with putting her down - my ds1 took against his cot when he was about 18 months and screamed blue murder if I put him in it. Your dd is a little young for a bed but that's what I did in the end with ds1 (sorry, you probably don't want to hear that!)
Usual after effects of MMR are a raised temperature, sometimes a rash, or swollen glands.
HTH.