Can you play the act 1 levels in any order?

The order isn't really matter here, since it dosen't effect in any way any of the stages so I believe it should be OK.

The only problem is remembering the stages you have already done during a run so I would recommend using live split

Yeah much like the Sonic Unleashed counterparts of All Day and All Night Stages, you can do them in whatever order you see fit as long as you do them all :)