To answer the other half of your question, things like squashes and courgettes are a good start as you can plant them through a layer of membrane or a thick mulch to help keep the grass down. Potatoes are good too, by the time you've planted them, earthed them up and dug them up to harvest, the bed will be in much better condition for next year! And have a bed for whatever veg you like best, the more you enjoy the produce at the end, the more worthwhile it will be getting there.
It might be worth using your space at home for starting things off to transplant to the plot too, gives a bit more time to get the allotment ready.
One warning about rotavating, it can cut up weed roots which then grow back from every bit, leaving it worse than before you started! Either dig by hand, spray with weedkiller then rotavate once things are dead, or use no-dig methods.