Are Hindi and Urdu (Hindustani) descendants of Sanskrit? I keep getting different answers from different places?

From what I know of these languages, this is the most accurate response. I recently got into a lengthy argument with people on r/India that "Urdu" in essence predates Hindi, because as you stated in your summary, this single language (whatever it may be called), came to preeminence during and after the Mughal period, where it was popularized under the name "Urdu". The "Urdu" name attempts to explain the language's origin, whereas the name "Hindi" simply means "Indian", which to me it's not clear why a dialect would be officially dubbed "the language of India" in name until after it had ascended to power and influence as Urdu clearly did from the Mughal period onward.

What I was told by opponents is that "Hindi" predates Urdu, and Urdu and Hindi were never the same language, which contradicts the widely accepted notion that they are in fact one language or at least diverged from one language. They will say that both diverged from pre-Islamic Khariboli, yet that doesn't explain all the Arabic and Persian loanwords present in Hindi which seems to be Urdu's hallmark. Wikipedia says "Hindi" was not an official language anywhere in India until 1881, and Hindi's orthography and grammar were not standardized until the 1950's, whereas Urdu had been famously known hundreds of years prior.

The argument just goes in circles and circles forever and never ends, as it's basically born out of animosity and name-gaming. Call it Hindi or Urdu, it is a rose that smells just as sweet.

With respect to OP's question, however, nobody questions that both Hindi and Urdu are based on Khariboli, which is certainly a descendant of Sanksrit.

I dont think the confusing part is the entirely subjective (and pointless) debate of what is "real" hindustani hindi vs urdu.

The confusing part is what did 9th-10th century proto-hindustani (Shauraseni) look like before the mughal invasion? The common understanding is that it was an evolution incorporating loan words from persian, but, at least to the layman, it seems by the time a modern hindustani emerges (13th century onwards) that it is so heavily influenced by persian both in grammatical structure and vocabulary that it's not clear if you can even say there is a single root language. (see my post below with some examples)

How did they come to this distinction. I'm looking for examples of the evolution from 9th>10th>11th>12th century (like we have for english), but I can't seem to find anything

Where do English words fit on this scale? For example, the Sanskrit "pustak", Persian "kitaab" and English "buk" all used to mean "book".

Indo-Aryan linguistics, the Sanskrit terms "tadbhava" (born from that), i.e. words that have evolved from Sanskrit through various stages over time, and "tatsama" (same as that), direct borrowing from Sanskrit, are useful in describing this.

Or in the tradition of historical linguistics more broadly, "inherited" words versus "borrowed" words.

Why does this seem to be much less stated outright than the case with Latin and Romance languages?

I am also not a linguist, but there are some glaring inaccuracies in your two points regarding grammatical similarities.

The infinitive in Hindustani ends in -na, but in -dan in Persian. Similar to the Hindustani termination is the Punjabi -na (with retroflex n) and Nepali -nu. I'd assume this ending is an innovation from the Sauraseni Prakrit stage. The Eastern zone languages (Bengali, Oriya, Assamese, etc) do not have this so it might not have been a thing in the Magadhi Prakrit, from which they descend.

As with many Indo-European languages, the verb 'to be' is a complicated one, and often resulted in several Proto-IE verbs coalescing into fewer or one copula in descendant languages. The Persian verb has the infinitive root "bud" or "bash", from PIE *bʰuH- (also source of English "be"), with conjugated forms like ast/hast from PIE *h₁es-. The Hindustani infinitive root "ho-" also comes from *bʰuH-, via Sanskrit "bhavati" and Middle Indo-Aryan "ho-" (this sound change seems to have taken place throughout the subcontinent from Punjabi to Oriya), not from Sanskrit "asti" from PIE *h₁es- like the Persian "hast". The conjugated forms like hum, hai, etc in Hindustani come from *h₁es- via MIA aha-. Both Persian and Hindustani have "to be" verbs that have forms from several predecessor verbs but it doesn't prove Persian had such a strong influence on Hindustani grammar. They seem to have developed independently, resulting in somewhat similar forms.

Bengali has a much more complicated copula system. The "ho-" root from Sanskrit "bhavati" is a completely independent verb, having a complete conjugation pattern and a connotation that's more like "become" rather than simply "be". Sanskrit "asti" became the defective verb "ach-" (from the metathesis of the st in the root) that only has present and past forms. Unlike Hindi, they did not coalesce into one single verb that looks like it could be from Persian.

The fact that both Persian and Hindustani use the copula extensively as an auxiliary in verb conjugations (as well as compound verbs in general) is probably more of a general areal feature, rather than strong Persian influence on Hindustani, and it is probably an innovation from the medieval period onward. Even Oriya's verb forms exhibit such behavior and that language has had the least influence of Persian among the major IA languages.

In general Persian grammar is much more analytical than Hindustani, relying more heavily on clitics. The only glaring grammatical influences from Persian, besides phrases borrowed wholesale ("khush amadid" - "welcome") I can think of in Hindustani is the ezafe which is used primarily in "Urdu" literature, poetry, and formal speech and rarely in "Hindi", though most speakers are familiar enough with it to understand it. It isn't really productive in everyday Hindustani speech; and speakers would use the default genitive construction using the particle ka.

That's very interesting. But English is still a direct descendant of Proto-Germanic despite having the same amount of Romance words from the French invasion. I didn't realize the Persian language had changed Hindustani that much. I wonder if it was a more drastic change than the one English underwent under the Normans.