The Queen doesn't choose who gets most of the honours per se. The lists emerge from various arms of Government and collated by the Ceremonial Secretariat (part of the Cabinet Office) so technically the PM will make the decision which is ratified by the Queen.
Some grades of honour are chosen by the Queen, so if she is a F1 nut then he might join the Order of the Garter.
It is possible to send off nominations - the form is available with a bit of googling. It would be deserving, bearing in mind Alex Ferguson has already got one, and Ross Brawn also took the risk to keep the team afloat.
Probably just watches the podium ceremony whenever there's a chance she can sing "god save me". That joke never gets old for her apparently.
Also, according to some articles that came up when Hamilton got some award from her earlier this year she saw the last laps of last year's Interlagos grand prix.