If I start a new coop game can I earn achievements like I would in a singleplayer game? Is that progress shared between all players?
I can personally say that i've got steam achievements whilst in a multiplayer session I wasn't hosting. I don't believe they are tracked between all players though.
I got the achievement for catching a certain number of types of fish.
You are able to get "personal" achievments like catching fish, but the host gets achievemnts like upgrade the house or total money earned.
I've been able to confirm that achievements are unlocked for both the host and any farm hands. A friend and I were able to unlock the "Cowpoke - Earn 50,000g" achievement while playing a co-op farm.
In my experience, only the host can get achievements in a co-op game. The other players will not get achievements; but I am unsure if their actions will trigger acheivements for the host.