They are unlikely to let you walk away with the whole V5 as you could then not post it for a few weeks and clock up a few speeding tickets. You will be fine with just the bit you need off the bottom, just make sure it's all signed etc. And yes, as mentioned, MOT if it needs one and any service history paperwork bills etc. As for the private plate, I think that has to be done at a local DVLA office unless you do it afterwards by post which is very quick anyway. Good luck!