No a AT&T 6/6 Plus unlocked will Not work on Sprint. Period. However AT&T 6/6 Plus unlocked will work on Verizon. Provide that you insert an active Verizon sim card into the phone it will work just fine.
Yes, the Verizon iPhone 5 comes unlocked for GSM use out of the box. I've been using a Verizon 5 since launch a couple of years ago on my AT&T account. Granted, I am limited to HSPA+ (AT&T's 4G), but I've had no noticeable problems with using it on the network. I receive texts and calls just fine. Sent from my iPad.
Cfairbank - As much as I believe in the more discussion the better it is, it becomes a hassle when people hijack the thread. Now instead of focussing on question of phone from Verizon to ATT, your action is going to get the discussion on another issue of AT&T to Sprint. A separate question would have been better, in my opinion.
Verizon 5 will work on AT&T and you'll get HSPA+ only, no LTE. AT&T uses different LTE bands than Verizon and the iPhone didn't support multiple LTE bands until the 5S.
Verizon 5 will work on T-mobile but you'll get EDGE only. No 3G/4G, and No LTE. T mobile uses different GSM bands than A&T (in most markets) and deferent LTE bands than AT&T and Verizon. There are a few markets where T-mobile uses the same GSM bands as AT&T, but I have not seen a map/document that states where those areas are. This is a result of the spectrum that AT&T had to give T-mobile after the failed AT&T buyout.
I've been using my Verizon 5 on AT&T since Jan 2014.