Ras has to be assigned to Tri-City if he does not stay with the Wings (if the Wings decide to even make this happen) at any time of the season. I believe Ras can go back and forth between the Wings and Tri-City if needed. When the Wings season is done, the Wings (if they want to) can assign Ras to Tri-City for their playoff run and he stays there until the Americans season is over. If the Griffins are still in playoffs after the Americans are done, then Ras can be sent to GR on an ATO contract and be on the Griffins playoff roster. That is how I understand this.
If he's under contract with the Red Wings, I don't think an ATO is needed. That would be for guys that don't have a pro deal yet, or have signed an NHL contract that doesn't start till the next season (IE when Larkin played in GR).
Also, as per the CHL/NHL agreement, if Rasmussen is sent to Tri-City, he can't be re-called unless it's a "3rd emergency condition", which basically means the Wings have so many forwards out, they've already recalled 2 forwards on emergency from the AHL/ECHL. There's also a rule that says NHL teams can no longer emergency re-call CHL players after March 1st.
It seems like one exception to a paper up and down at the deadline is a player returning from a loan to another league after their teams season/playoffs are done. This is how Sulak became eligible after the deadline last season.
I'm guessing that Rasmussen would have to be assigned back to Tri-City at some point in order to be eligible for the AHL playoffs, and then he can join GR once Tri-City's season is over. However, I can't seem to find any indication that he needs to be assigned at a certain point in order to be eligible.