to 1.): You have 6 places to fill out of 6 persons. For the first place you have 6 possibilities, for the second place you have 5 possibilities, ... in the end you have totally 6*5*...*1 = 6! = 720 possibilities.
That means your answer was correct.
to 2.): First you have to decide which gender starts the line: 2 possibilities. For the first place you have 3 possibilities, for the second place you have 3 possibilities, for the third place you have 2 possibilities, ... in the end you have totally 2 * 3! * 3! = 72 possibilities
to 3.): You have 4 places to fill out of 4 persons. According to the considerations of #1.) you have totally 4! = 24 possibilities.
to 4.): There are 2 different cases:
a) A takes the first place then you have 4 possibilities to place D. The same situation occurs if A takes the last place. In total: 2 * 4 = 8 possibilities.
b) A take the second to 5th place then you have 3 possibilities to place D. In total: 4 * 3 = 12 possibilities
There are 20 possibilities so that A and D don't stand together.