It sounds like you could not plan further in advance, but you are going to find certain issues that you will need to address before you commit to anything.
A minivan here is a commercial vehicle, with no windows, which is presumably not what you want to hire? I presume that you need a people carrier. A minibus is larger again and will seat say 15 people. Most cars here are manual and you may struggle at such short notice to get an automatic. I would not recommend hiring a car in the UK if it is going to be driven mostly on the wrong side of the road in mainland Europe. Insurance will be expensive to take it abroad too.
Day 1 - you will be jet lagged and will need to take it easy and recover. Spend the day in Kent - plenty of castles, gardens, shops etc that should suit the whole family. You could spend a whole week just in Kent! I am biased because I live there but there is lots to see.
Day 2 - It sounds like you are staying in Kent, so Bath and Stonehenge is a very long drive if you are still a bit jet lagged. Go up to London instead.
Day 3 - go to Rye or Lewes or Brighton, as the Cotswolds from Kent will be a minimum 3 hour drive in each direction and more likely 4 hours with traffic.
Personally, I would spend a bit longer in the UK.
To get to the continent, you have several options: fly London return to Paris is likely to cost upwards of £1,000 return for 5 even on Easyjet plus the cost of getting to the airport; Eurostar to Paris - all the cheap tickets have gone, so your dates are showing just over £1,000 for all 5; ferry or Eurotunnel, which will be cheaper to cross but you have the car hire price and insurance to consider. There have been a number of weekly strikes in France which have closed the ferries for several days at a time and caused huge delays to haulage and holidaymakers - the latest was just a few days ago and the issue is still ongoing (google Operation Stack to see the impact).
If travelling on the continent, you would be better hiring a car over there, but this will need some research on your part for costs/availability etc.
If you want to spend time other than just sitting in a car, I would suggest that you limit yourself to the UK, Paris area and Belgium/Holland. Road congestion here is far, far worse than anything in the States (I have driven in San Fran, LA, NY and Washington), so allow plenty of time for your journeys, despite what online websites may quote you. You do not want to be driving in Milan or Paris. School holidays are now in full swing in England, so accommodation will be more difficult to find, but there will still be accommodation available. Most hotels will not permit 5 in a room, so you will probably have to take 2 rooms. Alternatively, for a different holiday than you have suggested, look to stay in the UK and find a last minute cottage rental for a week then do a few days in Paris. Good luck.