According to Mass Effect Wiki, you unlock this achievement by completing 45 assignments with Wrex out of the disc's 73. Each DLC adds 3 to that list, making for a max of 79.
1) You do not need the character in your party to count toward your total, you just need him there when the assignment is marked "completed"
2) YES, DLC assignments can count, and are much easier than some in game assignments.
3) It make little difference who you use as your other character, as all the other characters need 45 assignments complete, save Liara who needs 50 assignments. Pick the squadmate who seems ideal for the situation or who will compliment your squad best.
4) When you earn this, all characters will regenerate 1HP per second no matter what.