Maximize "coverage" or maximize speeds and throughput?
If it's only coverage you're concerned about, then so long as you have whatever is sitting in your family room set up as an AP, you should get decent coverage there.
If you're interested in highest speeds and throughput, then you'd be well served (IMHO) to get rid of the MI24WR completely (it's an Actiontec 802.11g, and clearly is obsolete if you have any 802.11n or ac devices (e.g., modern smartphones, tablets, newer laptops, etc.).
I would use the G1100 so it connects first, and then using one of it's GigE ports to do your 125' run directly to your family room, where I'd connect something newer than the MI24WR...e.g., either your E3200 (which is an 802.11a/b/g/n only device), or better yet, I'd buy an AC1750 or AC1900 capable device that will provide better wireless coverage and wireless speed and throughput with 802.11ac speeds (of course that will mean that perhaps you'll have to upgrade some wireless adapters in devices, but most new smartphones use 802.11ac, e.g., iPhone 5's and the Galaxy S3's, 4's or 5's, as do most new tablet devices and newer laptops).
Of course, this may in part be a trick question...because you're running 125' of CAT6 cable, that may be an issue with signal strength (I don't know what the maximum cable runs can be before too much signal loss). [Edit to add: Never mind...just did some quick looking and see that the maximum runs can be 328 ft, or 100 meters].
But since the new G1100 also has MOCA, you could even eliminate the CAT6 cable run and just use a MOCA adapter to connect to your family room (assuming you have coax cabling in your house) where you could then connect another MOCA adapter to whatever new AP you're going to connect there.
In short, use the G1100 as the router, and as first device to connect to the FIOS ONT, then use MOCA to connect to the Access Point (802.11ac dual channel device) in your family room.
One other thought. Try using just the G1100; it's an AC1750 device and you may find that it provides sufficient wireless coverage in your family room even without a separate access point. Just a thought....
Lastly, the G1100 looks pretty interesting. Let us know how it performs and whether you're actually intending to use the Z-Wave/Zigbee capability (personally, I think that's just a recipe for more interference and insecurity...but who knows... this use of a combo router/wireless to do Home-based IoT is really too new to know).