It was not a country, but the area ruled by Mughals was called by Hindustan (coming from Hind,) particularly Muslim majority cities like Dilli, Agra, Lucknow,etc. They were almost completely Perso-Turkish in culture.
As states earlier, Muslim civilization was wiped from that area and Hindus were settled on those lands.
I am from Faisalabad, and yes, we love our stage shows and comedy. Faisalabadi Punjabi is unique in all of Pakistan and I believe, the most hilarious dialect. Most funny Punjabi programs are spoken in that dialect.
As a THetH Punjabi speaker with Pukhto speaking friends and relatives, I can tell you that if you learn to differentiate the accent, you can pick of most of what is being said. The same applies to Farsi/Dari and to a lesser extent Turkish.
Dari should have become the national language of Pakistan, but two mistakes forced our hand, namely Bangladesh area and influx of Muhajirs who wanted their tongue as the national language.
It is never too late to correct this mistake and I believe Pakistan has been afforded a great opportunity with IK, PTI, and KPK now in the spotlight in Pakistan. We can work on changing Urdu by bringing more Dari/Pukhto words slowly.
I also don't enjoy Punjabi unless it is pure and has Farsi, Arabic words, it gives so much culture and class to a language to have larger ideas (mostly taken from poetry and the Quran) in our lingo. I am a fan of Mushairi for this reason.
Our cultural artifacts, textiles, pottery, and woven crafts are mainly from our nomadic steppe Iranic heritage. It is similar to those of Turkic people, as well.