Once all of the surveys, searches have been done and everything has been agreed, you will sign papers in readiness for your conveyancer to exchange them with your buyer and your vendor. At the same point, all parties agree a completion date (which is the date you pay the funds and get your keys). Ideally, all parties in the chain complete on the same date.
Once you've exchanged, you'll have a completion date after which you can start planning your renovations. Don't book anything before you've exchanged because it might not all go to plan.
Your conveyancer and estate agent should be able to talk you through all of this and guide you on what steps you need to be planning for.
Yes, you leave your house the same day you get the keys to your new place. It's a juggle and lots of waiting around following by frantic unloading of the removal van at your new place. Try to get the kids looked after for the day.
Once you're in your new place, you can start painting. In my experience, I've done renovations and/decorating bit by bit which is the more costly way to do it but most effective way to be able to live in the house while the works are done.
If you have the funds, you could get a short term rent to live in while your new home is being renovated, but renovations often over-run so be prepared for extra rental and build costs.