New to MFT and need lens advice

Well, if you've already got the 12-60, the next lens I think you should get for telephoto is the Panny 100-300..... good deals available used if you shop around. I've got one and for what it is, it's IMHO excellent. I've used it on Olympus OM-D bodies, have taken a few hand-held shots of the moon and am amazed at the sharpness (hand-held).

(If money is no object, go for the 100-400).

In general though I have found that Panasonic lenses work best on Lumix bodies, and Olympus lenses work best on Olympus bodies.  If you're looking for a wide-angle, the Lumix 7-14 is great in many ways but if you don't know what chromatic aberration is now, you sure will find out using this lens!

Being new to MFT I have a few questions:

1. Is it worth getting a 43-MFT adapter to use my Oly FT lenses on the GX7? If so, any recommendations?

for 20 dollars you get more lens options

1. Is there a benefit to sticking with Panasonic MFT glass on a Panasonic camera or do Oly (and other manufacturers) lenses work just fine?

There are benefits, so get at least one panasonic lens with a panasonic body.

1. 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?

The panasonic 12-60 3.5 is a good walkaround lens, and accurate.

1. 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?

I like both the Olympus 40-150 for this, but get other suggestions on this one.

I had the E300 and E510, with both versions of the 14-45 and 50-150 zooms, plus the 70-300 zoom, when I first tried out M43 in 2010 with the EPL1.

I used the smaller FT 40-150 zoom that first year, with a Panasonic DMW-MA1. In fact, Olympus marketed that zoom with their adapter for M43 for a while, prior to the release of the M43 version, There was a firmware upgrade for both it and the 70-300 to mitigate focus hunt. I still use the 70-300 today with the adapter. Another lens I picked up was the FT 35mm macro, Quite a slow focuser. Never had the privilege of shooting the 14-54 or the 50-150.

Later, I bought a Viltrox JY-43F adapter for about $39, It works as well as the $120 Panasonic MA1. I didn't really look to see what SLR lenses you own. I certainly would not buy any FT lenses, but the Viltrox adaoter is an inexpensive way of using some of your old ones.

Good and not too expensive sharp lenses to complement the 12-32 are the Olympus 45mm f1.8, the plastic Olympus 40-160 zoom, Panasonic 20mm.

Going from 4/3 to m4/3 I thought the sensible thing to do was to get an adapter to be able to continue to use my old lenses.  I soon found that a hassle and wondered what I was missing in not having m4/3 lenses so I sold all my 4/3 gear.  I do miss the 50mm f2 macro though but otherwise I'm much happier.
So unless you have a particularly special 4/3 lens that you intend to still get a lot of use out of, I'd recommend making a complete switch.

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.