Lots of puzzles here. I wouldn't think the Sprint <?> T-Mobile issue is a factor either, that's a cellular connectivity thing as opposed to WiFi. Anyway, that SBG6580 appears to be a pretty good modem/router unit and since your stable of phones are also apparently in working order (they work fine using connecting to other WiFi networks) it really does seem to be a matter of 'there's something going on with your network'. Really, as you've gone through things already the fact that just your Note 3 can detect and connect to your home WiFi network is an indicator that there's likely to be some kind of configuration setting that your newer devices are balking at.
https://www.arris.com/surfboard/products/wi-fi-cable-modems/sbg6580-2/
So you could go into your modem/router setup menu, dig through the multitude of options and try to determine which one is the source of the problem. Sometimes that's a really good way to learn what your router is capable of too. If you don't already have it, here's a link to the pdf users' manual:
http://arris.force.com/consumers/Co...&c=SURFboard Cable Modems and Gateways#panel3
I'm going to suggest you set up a completely new WiFi network, if you're currently relying on the default SSID and password, set up a new SSID and new password. You might also want to avoid using the 'wizard' auto-config option and manually enter your own settings, those 'wizards' tend to fall back on defaults established to suit your ISP using generic options. You should be setting up your home network to suit your needs, not theirs.
This RouterCheck app is a good guide in setting up basic security in your router. Configure your new network and hopefully your S8P, Note 5, and Note 3 will all be detecting and connecting to it, then install this app on one of them. Do a scan and it will flag items it finds as possible vulnerabilities. What I really like about this app in particular is the documentation it includes, so you can read through not just a simple notation but details in simple terms on what it is, why it's a problem, and possible fixes in your router's settings.
https://play.google.com/store/apps/details?id=com.Sericon.RouterCheck.client.android&hl=en
Also, you might not want to pass on a 2.4GHz band network completely, 5GHz can provide faster bandwidth but 2.4GHz gives you more range. And don't get too crazy changing too many defaults in the initial setup, get your basic network working, and then tweak things later (keeping track of your changes so you just restore them if necessary).
