The wifi icon (click here to see how you could have found about it on your own) is the top bar, right between the carrier and the clock, and it replaces 3G or E. It looks like this:
Like Alvar said, best way to be sure is getting in Airplane mode and turning wifi on.
As for youtube not working while others do, it's probably blocked. This could be on the wifi or on the iPhone, if it's a company phone with a provisioning profiles blocking it or if it's jailbroken and got a firewall on it. Just check if youtube works on any other device on the same wifi and you'll know for sure where's the issue.
If youtube is just running slow, but it is running, it could also be your other heavy sites are cached. Without more info I can't think of more options for that issue.
If re-setting your network connections didn't help, it could be a hardware problem.
Depending on when it was made, it could be that your WiFi chip inside the phone has pulled (or, rather, is in the process of pulling) off of the circuit board. Late 4S models also had this problem, and here's how it manifests (progressively):
- Very slow speeds / drops connection to the network intermittently
- Stops seeing most (or all) networks, but the WiFi button still works, it's not grayed out
- Can't turn WiFi on at all, the button is grayed out in the settings menu - but the WiFi button in the control center might appear to work.
If your phone is under warranty, then take it to an Apple store and have them replace it. Believe it or not, the repair is quite easy, so if it's out of warranty and you know of a place that fixes phones, they've probably got some spare chips and won't charge too much. Apparently, it's a manufacturing defect, so a repaired phone fixed by someone that knows their stuff could last quite some time.
This has only been a problem with certain 4S and 5 models, you'll see a bunch of people reporting it if you search for something like "iPhone in freezer" - because (believe it or not), putting your phone in a Ziplock and then freezing it for a few minutes has been known to fix the problem for a short time (though I don't recommend doing this).
AFAIK, this isn't an issue with 5C, 5S or recently (as in the last year) manufactured iPhone 5 units. Could be a good opportunity to upgrade, even if you manage to get it fixed.
Best Answer
You have done many of the quick fixes, but the best guideline on troubleshooting this would be to re-start the troubleshooting process:
Before you start, do you have a backup of your device?
Also, you can test the data connection with some very simple items to rule out Mobile Safari or other items being blocked.
At that point, I would then go through the steps in the above article, noting on a piece of paper what results you get. That way should you need to contact AT&T or Apple - they can't give you the run around. It is Apple's responsibility to explain how to test the device up to and including restore / erase and setting it up new. It's AT&T's responsibility to ensure your data service is working if you are paying for it. You can not give them an easy out if you document your efforts to solve it and have done all the things that are expected of the end user.
Hopefully something simple like a power cycle and eliminating the WiFi will allow the cellular data to start working again, but it still could be something out of your control and you'll need help from Apple or AT&T to finally solve it.