What is the etymology of "oil"? Is the cute description of oil and petrol in the poem Autogeddon fiction?

No, there doesn't seem to be any evidence suggesting that oil/petrol/petroleum came from Sanskrit in any way. All three of these words have different etymologies, but are ultimately related to Latin oleum.

Etymology of oil

A simple etymology is given in Oxford Dictionaries:

Middle English: from Old Northern French olie, Old French oile, from Latin oleum ‘(olive) oil’; compare with olea ‘olive.’

OED agrees with this etymology, but is significantly more detailed. For example, it attests to over 100 different spellings of the word over the years! Many of the earlier spellings support the link with French, especially the ones spelled with ui. Others, especially ones that are spelled with eo (e.g. eoli) are thought to be related to Old English ele (which also came from Latin oleum). In any case, there's no evidence that Sanskrit had any part in the etymology.

The freely-available Middle English Dictionary has examples of sentences with a number of the spellings that were found in ME.

Etymology of petroleum

Petroleum comes from Latin petroleum, which is from petra (rock) and oleum (oil). (This etymology can be found in both the OED and Oxford Dictionaries).

Etymology of petrol

According to Oxford Dictionaries, the etymology of petrol is:

Late 19th century: from French pétrole, from medieval Latin petroleum (see petroleum).

Etymology of Latin oleum

According to Wiktionary:

Oleum:

From Ancient Greek ἔλαιον (élaion, “olive oil”).

ἔλαιον:

From Proto-Hellenic *élaiwon, identical to the modern Cypriot form. Compare ἐλαία (elaía, “olive”).

The ἔλαιον root is also supported by this Latin etymology book from 1828.