Only use apostrophes in contractions, and to show possession
Apostrophe has only one name. It's apostrophe. The plural is apostrophes.
Yes
That is the correct spelling of "apostrophe" (the punctuation mark ' ).
imbued in you
That is the correct spelling of the word "diaphragm" in all usages.
He was researching where the correct place to put apostrophes is.
Plurals don't use apostrophes, so the first one is correct.
sixth
pros and cons is the correct noun
Both usages are correct, as in these examples: "A circle is round, so it is drawn round." "I drew a circle around the correct answer."
The correct placement of apostrophes in your sentence would be: "Mike's brother's friend's cousin did a brilliant project on Saturn's moons." The apostrophes indicate possession for Mike (his brother), his brother (his friend), and the moons belonging to Saturn.
Grammatically, Presidents' Day is NOT correct. The correct spelling should have no apostrophes at all as the day does not BELONG to the presidents, therefore it is not possessive.