All year is generally the same, we don't have any snowfall, maybe two days in January. It is never too cold or never too hot.

If you want to see only london it should take around 3 days.

Yes it would be a waste to rent a car because in city it cost a lot of money to park and it is sometimes hard to find a car park, also that the underground is very good, also we have private buses that you can get from most of the bus stops in central london, they are especillay designed for tourism, they cost around 15 pounds a day and they go around the city and the major sites.
like:
http://www.bigbus.co.uk/
http://www.theoriginaltour.com/

Don't book tickets online you can buy them on the same day. For underground one day travelcard is around 5 pounds.

April march will not be very cold still you would need a thin jacket.

One thing you should know, it is going to rain, don't matter when you come, it is going to rain.

Things you must see:

The Houses of Parliament - Big ben

Madame Tussauds http://www.madame-tussauds.com/

Buckingham Palace: Queens Residence

Tower Bridge: A famous bridge

Harrods - Queen does her her shopping here, just go and have a look, very overly priced.

Oxford Circus - it is not a circus, it the middle of london and very good for shopping.

Piccadilly Circus - just famous and nearby china town.

London Eye: A big wheel, shows all of london, good at nights.

http://www.londontown.com/ for more info.