Indian consulate in the US will require the passport to be valid for at least for six months before issuing a visa (since most of their visas are valid for 6 months).
Once you have that visa stamp, then you can enter India as long as your US passport and Indian visa dates have not expired. E.g. you can take the visa six months before expiry, and enter India five months later and return to the US before the passport expires.
If you are staying in India for a long enough duration (several weeks), you can renew your passport here through local US consulate in most metros.
Contrary to Indian consulate in the US, the US consulate in India is a breeze to deal with - at least in Chennai. I once added pages to my US passport here. I was out in 20 minutes. There was just one other person in line there.
One of the things most of us find out at the last minute is that kids passports have ony a five year validity. This usually crops up as we plan for a prepaid holiday trip.
In recent months, the US State Dept is issuing paspsort cards (the size of a credit card), which can be used in lieu of passport when you travel to Canada, Mexico etc. For now, this is not meant for air travel. However, I think it is almost a certainty that they will allow you to use it in the future for any country. So it might be a good idea to get a card and a passport book when you renew next time.
Thanks and Regards