Has the intended driver ever driven in Central London before? I ask as some people get on ok with it, others really don't. It can be very slow going with solid traffic in parts. If I was going to Hastings from central London I would want to get the train. You can get trains from Victoria that take 2 hours or from Charing Cross just over an hour and a half.
Train would be the first choice - where do you intend picking up this car?
Train route is probably just as scenic as a car route and you (the driver) can actually look around at the scenery rather than just watching the back of the vehicle in front
The trains from Charing Cross to Hastings stop at Waterloo East and London Bridge - both very convenient stations for Bankside.
Picking up a car in central London would be my idea of purgatory. Hideous traffic and (if weekday) the hassle / cost of dealing with the congestion charge.
If you need a car for other reasons, take a train to Gatwick and pick up at Gatwick. Quicker,less hassle and probably cheaper.
I would also suggest forgetting the idea of renting a car. Hastings is easily accessible from London Charing Cross/Waterloo East/London Bridge by train in around 90 minutes. Places nearby of interest such as Rye and Battle are just a short train ride away. Imo, renting a car would be pita for many reasons - Driving out of London, the A21 to Hastings which south of Tunbridge Wells is known as the snail trail for a reason, plus the hassle of parking up in Hastings. For what reason do you wish to visit Dover? Aside from the castle, it really is rather 'meh.' A not overly attractive town for exiting and entering the UK from France.
As others have said train would be quicker and more scenic than car from London to Hastings. If you wanted to break up the journey you could always get off the train at Tunbridge Wells or Battle of both (both pretty town's on the line down) a standard ordeal or ssuper offpeak return will allow you to break the journey i.e get off or on the train. You probably won't need a car in Hastings as is pretty walkable as long as you donations hills (though is a slight advantage for getting to the main part of the country park) and there are lots of nice nearby places easily accessible by train bus or both most notably Rye. If you wanted to visit Cover that too is possible by train (the route at the end runs right under the cliffs).