IF Cavalli doesn't appeal, then he get 1 start vs the NYY before the ASB. If Mikolas appeals, starts tomorrow or Sunday, then drops his appeal, he can get in 2 starts before the ASB (for better or worse - it's still Miles Mikolas).
Griffin gets today's start and Wednesday vs the Astros (7/8).
Alvarez last pitched Wednesday, so he can go Monday on regular rest (7/6) and potentially start Saturday or Sunday vs the NYY.
Littell's next turn is Saturday 7/4. He looks like next Friday's starter given Alvarez's and Griffin's availability being < 4 days rest and Mikolas and Cavalli likely to be still suspended.
Mikolas could pitch Saturday or Sunday (7/4 or 5) if he appeals and his case is not hear, then drop the appeal and be available for Saturday and/or Sunday vs the NYY.
Best way to ensure Cavalli gets a start before the ASB is for him not to appeal. Then he can go Saturday or Sunday vs the NYY. Alternatively, he can appeal and try to pitch Sunday, then drop it; doing that, he's available 2nd game after the ASB.
Next Tuesday looks like a bullpen game. IF Alvarez is held until Tuesday, the Monday is the bullpen game and Alvarez can be available 7/12.
How's this for a rotation to the ASB:
7/3 - Griffin
7/4 - Mikolas
7/5 -- Littell
7/6 - Alvarez
7/7 - Bullpen
7/8 - Griffin
7/10 - Litttell
7/11 - Cavalli
7/12 - Alvarez / Mikolas