>> neither X nor Y.

gmat6087 wrote:a) 1)I am neither fond of swimming nor flying. incorrect

>> Can't be correct. "fond of swimming" <> "flying"

gmat6087 wrote: 2)I am neither fond of swimming nor of flying correct

>> Completely Correct.

gmat6087 wrote: 3)I am neither fond of swimming nor fond of flying correct but wordy

>> Partly Correct - redundant.

There are two other options.
4) I am fond neither of swimming nor of flying - incorrect. "neither" should be before action verb but after auxiliary verb.

5) I am fond of neither swimming nor flying - incorrect, the same as 4)

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b)

gmat6087 wrote: 1)I studied hard but failed in the exam.correct

>> This one is correct. "NO comma" before "but", so "failed" can refers to the initial subject.

gmat6087 wrote: 2)i studied hard, but I failed in the exam.correct but wordier than 1

>> "Comma" requires usage of subject "I".

gmat6087 wrote: 3)I studied hard, however I failed in the exam.incorrect. I thought here however acted as conjunction. But the explanation was that howver... emax is run on

>> "however" is a conjunctive adverb. So, usage of "comma" is wrong here.

https://en.wikipedia.org/wiki/Conjunctive_adverb
https://en.wikipedia.org/wiki/Grammatical_conjunction

"The following rules are considered to be correct punctuation for conjunctive adverbs:
1) Use a semicolon or period before the conjunctive adverb to separate two independent clauses joined by a conjunctive adverb. A conjunctive adverb is not strong enough to join two independent clauses without the aid of a semicolon. A comma may alternatively be used if a conjunction ("and", "but", etc.) appears between the first clause and the conjunctive adverb.

2) Use a comma following the conjunctive adverb when it appears at the beginning of the second clause unless the adverb is one syllable.

Like other adverbs, conjunctive adverbs may move around in the clause (or sentence) in which they appear. When they appear at the end of the clause, they are preceded by a comma. If they appear in the middle of the clause, they are normally enclosed in commas, though this rule is not absolute and is not always applied to very short clauses.'"