These sentences fit into three categories of mistakes: Grammar (G), Redundance (R), or Vocabulary (V). G or R or V at the end of each faulty sentence indicates the type of mistake made in it.
Faulty: Between you and I, he is the most qualified for the post. G (After a preposition (between) use objective case pronoun.)
Correct: Between you and me, he is the most qualified for the post.
Faulty: The two employees were quarrelling with one another. G (Each other two, more than two use one another.)
Correct: The two employees were quarrelling with each other. OrThree of the employees were quarrelling with one another.
Faulty: It will have more people at the meeting tomorrow. G
Correct: There will be more people at the meeting tomorrow.
Faulty: Why do you insist on me playing the piano? G (Use possessive case before a gerund or verbal noun (playing).)
Correct: Why do you insist on my playing the piano?
Faulty: He was advised to seriously study the report. G
Correct: He was advised to study the report seriously. Or
He was advised seriously to study the report. Keep infinitives (to study, to run, to work) together, whenever possible.
Faulty: Neither he nor his sister should have swam across the bay. G
Correct: Neither he nor his sister should have swum across the bay. This is an irregular past participle.
Faulty: I shall do my endeavour best to assist you in this matter. R
Correct: I shall do my best to assist you in this matter.
Faulty: It was not me you saw at the dance last night. G (Use the subjective case pronoun after the verb "to be.")
Correct: It was not I you saw at the dance last night.
Faulty: You could of gone to the meeting if you were interested. V ("Of" is pronounced ov, so it is often confused with "have," as it sounds correct, but isn't.)
Correct: You could have gone to the meeting if you were interested.
Faulty: Excuse me, sir. Who are you looking for? G (Avoid ending a sentence with a preposition. The preposition can be brought earlier into the sentence and used before the object pronoun whom.)
Correct: Excuse me, sir, for whom are you looking?
Faulty: When you coming back from town, don't forget to bring me a sandwich. G (You, by itself, is a pronoun, not a verb–you are or you're is pronoun and verb.)
Correct: When you're coming back from town, don't forget to bring me a sandwich.
Faulty: There is no need to speak so loud on the phone. G (Use an adverb (not an adjective) to modify the verb, speak.)
Correct: There is no need to speak so loudly on the phone.
Faulty: I must tell you of the amount of times I called you and there was no answer. G (Use number of before countable nouns–times)
Correct: I must tell you of the number of times I called you and there was no answer.
Faulty: The reason why she cannot attend the meeting is because her husband fell ill suddenly. G
Correct: She cannot attend the meeting because her husband fell ill suddenly.
Faulty: Where are you working presently? V (Presently in Standard English means "soon, in a while.")
Correct: Where are you working at present?