I've been doing some research today because this question has piqued my interest.
Among other things, it appears that the 6 device limit is somewhat of a myth perpetrated from dealing with an older generation of routers.
Most current routers should have a theoretical limit of 255 devices, based on what I have found. Of course, actually connecting that many devices would slow the system down severely.
The more devices you have connected, the more chance there is of interference between the devices, so drop-offs may not indicate the router is overloaded.
In particular, if you have an older G device, it can force the router into G mode, and will slow everything else down and may cause a dropout when switching modes.
The dual band routers allow you to distribute your devices better.
I do believe the N750 would do what you need.
Netgear also has the R6100/R6200 models that support 802.11ac. It's a newer protocol, and is very fast.
I tend to favor Netgear. We use Netgear and Cisco routers, and the Ciscos are consistently slower on throughput, as well as having a relatively high failure rate.
One thing that seems odd is that your i-devices don't retain the password.
I'm not sure what's up with that.
As for the cable connection, now we are in my realm.
My day to day job is cleaning up cable signal.
You can relocate the modem to another room if:
The cable in your home is proper size(RG6). Many homes are wired with RG59 because the builders thought they could save a few bucks. RG59 is crap.
The connectors are good. Crimp connectors introduce a ton of noise. Even though you may have good levels, if there is enough noise, your cable internet connection will be compromised.
Reconfiguring splitters to get the proper level to any outlet in the house is easy.
Ideally, the line serving your modem should be a dedicated line directly from the primary splitter.
High demand devices, such as game consoles, or video streaming (think Netflix)are best with a physical connection.
A high, central location is best for wireless.
Sometimes you have to compromise. Bear in mind that interior walls on the first floor of an existing two story house are virtually impossible to access.
You'll probably want a tech to make sure the relocation is successful.
With luck, you'll get a good one.
If he asks questions about what you want, checks behind the wall plate, and uses a meter, you are probably in good hands.