New to MFT and need lens advice

Let's see. You asked:

1. Is it worth getting a 43-MFT adapter to use my Oly FT lenses on the GX7? If so, any recommendations? - I'll say probably not, because 4/3 lenses have slower focus motors, because at the time that was what could be done. Recent m43 lenses will be smaller, lighter, and will focus faster.
2. Is there a benefit to sticking with Panasonic MFT glass on a Panasonic camera or do Oly (and other manufacturers) lenses work just fine? - I have mostly Panasonic glass, but also the Sigma 60mm for m43, works fine. Olympus lenses definitely work on Panasonic bodies but many of their lenses lack stabilization. If the GX7 doesn't have in body stabilization, then you may prefer to look for a Panasonic lens with some version of OIS (Power OIS, etc)
3. Even though I have the 12-32, I'm considering the 12-60 as my "walk-around" lens. (I have used the Oly 12-60 on my 43 gear and really liked its range.) Any thoughts on which 12-60? - Really depends on how much you can invest.
4. Lastly, I'd like something longer than the 12-60 to use occasionally for wildlife and candid photos of my grandkids playing. I've been looking at the Lumix 45-175. Any thoughts on that lens or anything else I should consider that is in that general price range? - you might consider the 14-140mm lens. It's surprisingly good, especially for a 10x zoom, but not quite as good (nor as expensive) as the 12-60mm. The 14-140 might become your "walk around" lens when outdoors. I do recommend you also get a fast lens for indoor lighting without flash. Choices 15mm f/1.7, 25mm f/1.4, 42.5mm f/1.7 (very small but excellent). For reach beyond 140mm on a budget you might want to check out the 100-300mm Panasonic zoom. You probably won't need it often, but if you're going to shoot wildlife, 175mm may not be enough. The 14-140 and the 100-300mm cover from fairly wide angle to telephoto, with a little overlap. I think you will know when you need to mount the 100-300mm.