House is 3400 square feet so need decent range.
Odds are high that no single router will cover the complete house - not without dead spots, at least. Walls and ceilings/floors disrupt the signal, so it's not just a matter of a router's maximum range, really.
In terms of "Apex compatibility", there's not going to be a specific device that is better/worse for the Apex. The controller will register itself on the router (or the network, technically) the same as any other device - such as a laptop or cell phone - would do. The best way to connect your Apex is using a hardline. That is; you are able to follow wire(s) running from the controller all the way to your modem (and then, conceptually, out the house, etc.). This will give you the best chance of having a stable connection at all times.
One thing that's emerging on the
market now is "mesh networks". These operate much like a router and a bunch of repeaters, but have some different software involved that acts to make a more stable network out of the mesh (doing better at "handing off" a device from one node to another, for example). One example of this is the
Google Wifi. I'm running these in my house and don't have any dead spots that I've found. I still have my Apex hardwired, however - it's traffic demands are such that any wireless signal (even full strength) will likely result in outages. As it is, I still get "phantom disconnects" (reports of power loss that are quickly restored) from time to time...
