For example, if my friend hosts a game and I join it and we go through the entirety of Act I, will I have to beat Act I all over again once I leave the co-op session?
While in CO-OP.
So you gain more achievements playing as a group than solo. You don't lose nothing. But remember monsters get 75% more health for each player in the party.
About the quest progress.
You'll be able to start all quest beyond 4 and the first quest in ACT III. But You'll need to complete the quests 1, 2 and 3 from ACT III.
You won't have to redo the act. Every quest your character completes is completed both when playing with, and without, friends.