Details: I hold a Indian passport with a valid and unexpired US student visa (F1). I would be travelling from Los Angeles to Mumbai via a 18 hr layover at London Heathrow airport in December 2012.
1) You don't have to apply for a transit visa if you are eligible for a Transit Without Visa (TWOV) concession. The rules for TWOV are outlined here: http://www.ukba.homeoffice.gov.uk/sitecontent/documents/policyandlaw/carriers/ukvisarequirements.pdf
Indian citizens may not TWOV in general. However, it looks like you meet the following exception:
TWOV is a concession which does not apply to nationals of the countries shown above in red who need visas even when in direct airside transit unless they are in possession of one of the following documents:
(ab) a valid visa for entry to Australia, Canada, New Zealand or the United States of America and a valid airline ticket for travel via the United Kingdom as part of a journey from the country in respect of which the visa is held to another country or territory;
So you should be able to TWOV.
2) You can do sightseeing.