No, it is not.
First and foremost, there are many languages recorded long before the advent of Sanskrit, and many religions recorded long before the advent of Hinduism. The oldest surviving texts in Ancient Egyptian are from c. 3000 BCE, while the majority of the Rigveda (the oldest known Sanskrit text) was probably composed between 1500 and 1200 BCE. So that's a difference of a millennium and a half—and that's just between the oldest surviving written documents in Egyptian and the oldest oral traditions that were, much later, written down in Sanskrit! (While scholars think the Rigveda was composed sometime in the second millennium BCE, it wasn't written down until centuries later, and we don't have any actual artifacts with Sanskrit written on them from this period.)
Sanskrit is an Indo-European (sometimes called Indogermanic) language, which makes it a relative of English, Spanish, Russian, and many others. But Sanskrit isn't the ancestor of those languages, any more than the composers of the Vedas were the ancestors of all modern Europeans. Rather, Sanskrit and those other languages all share a common ancestor, called "Proto-Indo-European" (or "Proto-Indogermanic"), which was spoken somewhere between 4500 and 2500 BCE.
There are no surviving records of Proto-Indo-European (PIE for short), but scholars have been able to reconstruct it by comparing the languages that are attested, and working backward from there. There are some striking similarities, which make it clear that the languages are related. But:
- Sanskrit preserves some features that disappeared in other branches: for example, the injunctive is well-attested in Vedic Sanskrit, but is uncommon in Homer's Greek, and disappears entirely by Plato's time.
- And other branches preserve some features which have disappeared in Sanskrit: Hittite retains a phoneme that disappeared entirely in Sanskrit (but left plenty of traces showing that it must once have existed).
There are dozens and dozens more correspondences like these, where features disappeared in one branch but survived in another, or were innovated in one branch but not in another, and so on. So while it's clear that Ancient Greek, Sanskrit, Hittite, and so on are related, it's also clear that none is "mother" to the others: they're more like "siblings" or "cousins", with a common ancestor.
Do you think that there must be a common language for our ancestors[?]
Now this question is harder to answer.
The techniques I mentioned above, called the "comparative method", are really useful for reconstructing languages that must once have existed but aren't directly attested. But this method can only go back so far. Past a certain point, the comparative method just can't say anything particularly meaningful.
So while we know that there was an ancestor to all Indo-European languages, and an ancestor to all Afro-Asiatic languages ("Proto-Afro-Asiatic"), and an ancestor to all Sino-Tibetan languages ("Proto-Sino-Tibetan")…we can't really say anything definite about what came before those. They might all descend from an ancient sort of "Proto-World", or they might have all come about independently. There's just not enough evidence to say one way or the other.