If I remember correctly, you can live on ~10 USD/day in India (and Nepal) including everything (food, lodging, travel within the country (train/bus), possible gifts for friends). That is if you do no climbing/trekking. If you hire porters/donkeymen, need to buy supplies/rent equipment, this could go up to ~30 USD/day. Of course this also depends on how high your requirements are
You could easily spend a full month in Rajasthan.
As for 'ruins', in the Delhi/Mumbai area I would recommend Khajuraho, Ellora, Ajanta, Fatehpur Sikri, Orcha among others.
I recommend the LP Trekking in the Indian Himalaya book.
Manali, McLeod Ganj are probably good places to find fellow trekkers at that time of year.
MichaelJ's recommendation on visiting Varanasi and then taking a train+bus+jeep to the Nepali border is a good one. You would then arrive sort of midway between Kathmandu and Pokhara and could also visit Chitwan national park en route.
I enjoyed Goa/Kerala as well, but those are very far south from Bombay/Delhi by train.