South India is coolest in December and January and again during the monsoon in June, because of the cloud.
The north (Delhi, Rajasthan etc) is at its best in the period October to early April.
But be aware that overnight fog can mess up air travel into and through Delhi in Dec / Jan. So dont make your plans too tight; allow some slack.
We built the house without ACs as a matter of policy, and in the early days slept outside in the traditional way; but recently, with an increased flow of summer visitors, we have installed ACs in the bedrooms to give a good nights sleep inside as well.
During summer, the high ceilings, good insulation and ceiling fans make the public rooms agreeable by day and retain the feeling of being close to nature.
Sleeping out on a charpai under a mosquito net is still a good option and is much enjoyed by those prepared to try it. A seventy year old lady in April 05 said that "under the stars, she had just had the best nights sleep in her life."
Somewhere in India the weather will be attractive at all times of the year. Do plan your trip to be accompanied by the sort of weather you like rather than fighting against it.