That google maps route is not correct. There is no need to backtrack from Fish Creek Canyon - just continue to on Hwy 88 to Roosevelt Lake. At Roosevelt you can turn right to Globe and Salt River Canyon, or left to Payson.
Looks like Payson/Strawberry have some backtracking, so I opted the Show Low route.
I don't see how that is - what backtracking are you talking about? From Strawberry just head north to Flagstaff and Grand Canyon, through Clints Well and Happy Jack. Its actually more direct than the route through Show Low, and with the same backtracked sections (from GC back to Sedona).
As an alt to Show Low, a friend suggested the Wigwam Motel in Holbrook. -- any input here? Although you want me to eat lunch in Show Low, it seems impossible to me as the 3-4 hike in Cibecue happens before we get to Show Low.
Holbrook and Show Low are less than an hour apart - it doesn't matter where you stay (or where you eat lunch or dinner, although Show Low has better restaurants. BTW, I was thinking about dinner). You can see the Wigwam on the internet, and it pretty much is what it is - either that interests you or it doesn't. heck, after Holbrook its only another 40 minutes of dull interstate driving to Winslow, where you could stay at the La Posada, which is awesome.
Basically, if you go the Payson-Strawberry route, there is less driving overall, and your main attractions or activities will be Tonto Natural Bridge State Park, and the hike to Fossil Springs, both of which are superb. The Show Low route has Salt River Canyon and the Cibecue Falls hike, both of which are also superb. The next leg of the drive to Flagstaff and Grand Canyon is prettier from Strawberry (quite lovely actually).
As for where to stay, we've covered that ground pretty well...unless there is some really special property you want to enjoy like the La Posada or some cozy B&B in Pine or Globe, I'd base it more on what you want to be doing that day - none of the towns are nice or interesting enough themselves to plan an itinerary around.