“A clergyman and philosopher entered the hall together.” “A clergyman and philosopher” means one person who is both clergyman and philosopher. The article should be repeated. “A clergyman and a philosopher entered the hall together.”
“A red and white flag” means one flag of two colors. “A red and a white flag” means two flags, a red flag and a white flag. “A great and a good man has departed.” The verb has implies that only one man has departed, hence the sentence should be, “A great and good man has departed.”
“They sang the first and second verse,” should be, “They sang the first and the second verse.” “The literal and figurative meaning of words” should be, “The literal and the figurative meaning of words.”
“In framing of his sentences he was very exact,” should be, “In the framing,” etc., or, “In framing his sentences he was very exact.” “The masculine and feminine gender,” should be, “The masculine and the feminine gender.”
“After singing a hymn, Miss Willard made a stirring address.” If Miss Willard alone sang the hymn the sentence is correct. If the congregation sang the hymn the sentence should be, “After the singing of a hymn, Miss Willard made a stirring address.”
“He is but a poor writer at best.” Say, “at the best.” “He received but a thousand votes at most.” Say, “at the most.”
“John came day before yesterday.” Say, “the day before yesterday.”
Article redundant
“Shakespeare was a greater writer than an actor,” should be, “Shakespeare was a greater writer than actor.”
“This is the kind of a tree of which he was speaking,” should be, “This is the kind of tree,” etc. “What kind of a bird is this?” should be, “What kind of bird.”