Definitely wait for the upgrades, and buy whichever fits your budget when they are released. The iPad Pro 12" has a worse camera and older-spec screen because the 9.7" was released ~1/2 a year later with 6S-era chips - that's why you can have stuff like always-on "Hey Siri" with the 9.7" and not the 12". Once the updates come out at the next iPad event you should expect the 12" to be the premium option again.
I use my girlfriend's iPad Air 2 as my daily driver, since I gave her my macbook pro for working at home. I also keep my mom's iPad 2 running when I go home to visit, so I have perspective from tech that's significantly older and significantly newer than your iPad 3. Any current-gen iPad will be a *lot* faster than the iPad 3 - it was only slightly faster than the iPad 2 when it was released, and it was the iPad 4 and on that got the significant GPU/CPU increases to deal with the retina screen. multitasking is a lot better in iOS 10
than before, and with a decent keyboard you can do all right with productivity. Microsoft Office for iPad is perfectly fine for spot checking/updating work documents unless you're buried in excel formulas 24/7.
That said, I'm not sure an iPad pro would be much better unless you need the Apple Pencil or a larger screen, specifically. If Apple kills the iPad Airs and only has the 9.7/12 Pro, then it's mainly a matter of speakers (2 vs 4) and screen size.
Definitely stick with Verizon if you travel outside major metro areas - in my experience T-Mobile and Sprint lose LTE coverage really quickly once you leave city borders. When I lived in Texas it wasn't uncommon for my Tmob/Sprint friends to not even have 3G while I still had full-speed LTE on Verizon. That said, iPads are universally compatible with the major carriers so you could stop by a T-mobile store and try one of their postpaid/monthly plans to see how that suits you.