[Galaxy S 5 Q&A, Help & Troubleshooting] Airman1947:
Had to get a new router :(

Ok, so I finally fixed the problem...in a manner that I wish I didn't have to resort to, but it seemed like the only option.

Let me start by thanking Rukbat from Android Central. I posted this topic there too, and his response was very helpful. Here's a repost of it:

Talk to Verizon (the tech people in the back, not the sales associates) or the Time Warner people (same thing - tech people) and see if anyone has an old Linksys or D-Link wifi router you can borrow for a day. (I'm using a Linksys WRT54G at the moment, and in router years it's as old as George Washington's corpse.)

From years of experience, writing communications code, playing with my phone, routers - and looking at my calendar - I'm guessing a bug in KitKat that shows up only with some apps. My almost new Linksys EA4500 started showing this problem almost as soon as the phones updated to 4.4.2, but the apps all work fine on old routers.

If an old router solves the problem, you can get a refurb WRT54 (G or GS) for the $25 range on Amazon (or maybe TW has an old one they'll let you use with one of their modems).

The Facebook app works just fine on mobile data, AT&T hotspots, TMobile hotspots - but not on new Linksys (supposedly the best) routers, and I've heard reports of the same problem on the newer D-Link routers, also supposed to be top-of-the-line. (I actually left the app running for 12 hours once, just to see how long it would take to update. It's more than 12 hours, because it still hadn't done anything. But standing in the parking lot at a Sam's Club, with a barely usable AT&T hotsopt signal, it updates in a couple of seconds.) AT&T connected me to their in-store wifi (which is secured, so of course they deleted the connection after we checked it out) and Facebook updated immediately. Speedtest.com shows over 14mbps download and pings running from 15-50ms with my new router that doesn't work with facebook, so it's not the connection, it's not the wifi signal itself, it's either a problem with certain apps or a problem with the 4.4 wifi firmware. Play store downloads are slower on the new routers too, but not too bad. just noticeably slower. (If I set up a guest account on the router it works fine, though, so it's not a bad router. It's just a pain to have to sign into the guest account every so often.)

This is the reason systems analysts turn grey at a young age and have bald spots. Problems that don't seem to make sense, and people in a position to debug them who can't or won't. I can reproduce the problem at will (which is the difficult part of fixing a problem), but there's no one interested in fixing it.

I ended up just getting a different router, as he suggested. From what he's describing, it sounds like some routers just don't play nice with some newer phones, for no apparent reason at all, which is both discouraging and frustrating at the same time. And I've come to the conclusion that my router is one of them. After spending hours on the phone with technical support from Verizon, Time Warner, Arris and Samsung, no one could find a solution for me.

I will say that I did find some hints from other places online that tweaking certain router settings will fix this problem, but I think I kind of nipped myself in the **** by not doing my homework when I bought my Arris modem/router. This router is locked down pretty tight from changing certain settings. You can't disable IPv6, which some people say fixed the problem. Someone else suggested enabling IP tunneling, but I don't see this setting on my router. Somewhere else, I read that this is the same thing as Port Forwarding, but I really don't know what port you would have to forward to a phone. You can't even update the firmware on my router, as Arris only provides it to cable ISPs...and they won't push it to customer owned devices. :P I don't even know why they sell this model to the public if that's the case.

So anyway, I just bought a much cheaper Netgear router from Best Buy, and wouldn't you know it, it works great! I think that it even outperforms my Arris. I guess the moral of the story is be careful about what you buy.