The issues are official/government vs local business vs home, educated vs colloquial, first language vs street vs languages taught at school (and these issues have some overlap).
Indian English is a full fledged variety of English, whose differences are primarily in some vocabulary and turns of phrase and some pronunciation.
"English is one of the two official languages of the Union Government of India."
This means that government matters are written and conducted in English (along with Hindi). Each of the states of India (Punjab, Gujarat, Kerala, Uttar Pradesh, etc) have their own official languages usually the main local language (eg Gujarati in Gujarat), and maybe English and maybe Hindi.
"...only a few hundred thousand Indians, or less than 0.1% of the total population, have English as their first language."
First language is what you learn from one of your parents at home from birth
- Knowing English at all, whether at home, school, fluent, or just barely getting by:
"12.6% of Indians know English"
This does not seem like much, but it is spread out throughout the country, so as a traveller, you can get by very well conversing with English.
- As to perspective, English is more for the educated:
"While there is an assumption that English is readily available in India, available studies show that its usage is actually restricted to the elite, because of inadequate education to large parts of the Indian population."
English is the most popular foreign language taught in schools, and fluent English is a requirement for acceptance into Indian universities.
To your explicit questions, there is a very very tiny set of people who learn English as a first language (but that is probably in a bilingual situation with another Indian language, rather than having two parents that speak only English. So it is barely noticeable as a language learned at home from birth.
But English is heavily spoken, especially in the university and in government, by the richer, more educated set. For most of this set, they are fluent bilingual speakers of English given constant schooling starting early. And anyway, the great majority of the population is very familiar with English.
And for the TV quote:
As an educated Indian, Rajesh is fully fluent in English, or rather, educated Indian English, probably started learning very early, maybe even as a second language at home before he started school.
Sheldon's use of 'native' does sound a little weird. Fluent totally, but 'native'? It's arguable (with good cases for both sides).
Frankly, Rajesh's command of the concepts and language that they are discussing is far better in English than in his other language.
I think the writers are just trying to make it known that Rajesh actually probably as good at English as Sheldon. To your point, I don't think the dialog is great at managing all the nuances.