According to law, you must have your passport with you at all times. To check into a hotel you must show your passport to the hotel clerk and fill out a form with your passport and visa information. If the police ask to see your passport, you must show it to them. They have the right to ask you for it at any time, for no reason.
If you plan to stay in India more than the period for which your visa was issued, then you will have to extend your visa. You need four photos. Depending on the office you apply at and your personal circumstances, you may or may not get an extension. The official policy is that a tourist visa cannot be extended. Usually the only reason that you will get an extension is because you were, or are sick (you need a doctor's letter saying this is a fact), or you have some very special circumstances. Also, if you can not get a flight out of India you can often get a visa extension. You must show your airline ticket with a confirmed flight date on it, and then you will usually be issued a visa for the number of days until the flight. WARNING! If you plan to stay in India for more than six months, you must register at a Foreigners' Registration Office within two weeks of your arrival
Police 100 Fire 101 Ambulance 102 These are the numbers in Delhi, but they are usually the same in other places too.