Illogical Meaning Drill_Veritas

2. Jonathan Swift wrote a newly discovered letter in the same year that he published Gulliver’s Travels.

Here, "newly discovered" doesn't make sense in context. If Swift wrote the letter, he didn't discover it himself. It must have been hidden for a while, then discovered recently. The "newly" places the discovery in a different time period from the writing of it. Here is a correct version:

Researchers recently discovered a letter that Jonathan Swift wrote in the same year that he published Gulliver’s Travels.

3. So persistent were John’s inquires at Harvard Business School that he was eventually admitted, even with a GMAT score well below the average at HBS.

This was explained to be correct while I believe this has pronoun error as he cannot refer to possessive form of John.
the comma was not required according to me as it does introduces modifier error.

The "possessive poison" rule that you cite is a tricky one. Some grammarians argue that a pronoun should not be used to refer to a noun that was in the possessive form. GMAT SC questions will sometimes fix this issue. However, there are examples of real OG questions that don't adhere to this rule, for example:

OG #109

Among the objects found in the excavated temple were small terra-cotta effigies left by supplicants who were either asking the goddess Bona Dea's aid in healing physical and mental ills or thanking her for such help.

(A) in healing physical and mental ills or thanking her for such help
(B) in healing physical and mental ills and to thank her for helping
(C) in healing physical and mental ills, and thanking her for helping
(D) to heal physical and mental ills or to thank her for such help
(E) to heal physical and mental ills or thanking her for such help

Here, the correct answer was A. You'll notice, though, that all of the answer choices use "her" to refer to Bona Dea.

As for your question about commas... the short answer is: don't worry about it! The GMAT will not test you on comma placement. An answer will never be determined right or wrong based on whether a comma should be there or not.

For more on possessive poison, see here: https://www.beatthegmat.com/mgmat-awkwar ... tml#695917