I did this spectacular trip way back in 1987 and have been longing to do it again! For details, check Doug Lorain's "Backpacking Washington" and numerous trip reports on www.nwhikers.net. I haven't seen any reports for the loop this year, though, so check reports for the individual destinations. No, you don't cross the Suiattle unless you want to go around the west side of Glacier Peak, a far longer trip!
The one part that is not trail is along the level "Spider Glacier" (really a snowfield, not a glacier) to Spider Gap, no big deal, and then the descent of the snowfield to upper Lyman Lake. If the snowfield is hard or icy you're in trouble unless you have an ice axe and know how to use it. If it's mushy, which it will be by mid-afternoon unless it's cold, you can easily get down plunge stepping or (if you're more adventurous than me) doing a standing glissade.
I'd time your entrance or exit on a weekend so you can get a ride between the two trailheads. I went with a group, so we had two cars.