Ok with no responses here so I dug deeper.
At discussions.apple.com users said YES for the iP5s but NOT the iP5.
Same here:
https://www.whistleout.com/CellPhones/Guides/Taking-your-Verizon-iPhone-to-another-carrier
Still don't know about the APN settings but will just have to try and see.
With an iP6 I was able to simply insert a TMO sim.
Your iPhone 5 is the A1429. That's the Sprint/Verizon model.
I can confirm that it will work on T-Mobile. I have two iPhone 5's (5 not 5s) on T-Mobile.
However, keep in mind that you will NOT get LTE. The best they can do is 4G. In my area 4G does around 12-13mbps down so not really a problem for me.
Also, another thing to keep in mind is that as T-Mobile refarms their spectrum 4G for the iPhone 5 will stop working sooner or later.
My particular two iPhone 5's are Sprint models. Both were replaced in early 2017 and Apple allowed me to choose my carrier (because I was no longer on Sprint). I chose Verizon both times because LTE Verizon phones (at least back then) were unlocked. So both old Sprint iPhones fully unlocked and too my T-Mobile ID.
And that's all you need to do is insert a T-Mobile SIM and have it activated. You don't need to fool with APNs.
Finally, you may need to contact T-Mob about voicemail. I've never gotten it to work on my 5, but I use that phone as a secondary phone so I've just never gotten around to talking to tech support about it.
https://forums.macrumors.com/threads/sprizon.2036201/