Driving in the UK and mainland Europe is perfectly traightforward - a couple of hundred million of us manage every day
The advice about booking an automatic is good - they are not rare but neither are they standard and rental companies assume that, if you've booked one, you might not be able to drive a manual.
If you do hire a car, make sure you return it in the same country you collect iut - otherwise it get horrendously expensive. Also, it doesn't make sense to take a UK car out of the UK or a continental car to the UK. Don't try to run a car in London - parking and congestion charges make it a crazy option.
My wife gritted her teeth and took our two children (7 & 12) to Disneyland Paris at Easter - she was very impressed by the hotel. I'll try to find which one it was.
London is likely to be horrendous during the Olympics, you really should think about whether you want to get involved. Croydon, by the way, is probably the dreariest suburb in the UK - possibly a contender at a European level... I can't imagine anything worse than being there then - most Londoners are planning to leave
As others have suggested, especially with children, I would urge you not to try to fit in too much. Although it looks small, travel is typically slower in Europe and you risk spending more time travelling than enjoying. With a few esceptions - the Alps, parts of Scotland, for example - there will be little pleasure in the driving. Mix trains, planes & automobiles judiciously and you will still be a ble to see a great deal in the time you have. Try the overnight train from Paris to Venice, for example.
Although my home city is now Glasgow, I have to say you should go to Edinburgh as a base for visiting Scotland. It's a genuinely stunning city with much to see and do. On the outskirts, in the old port of Leith, the retired Royal Yacht Britannia is well worth a visit.