How hot can you let a barrel get safely?  Is there a certain number of rounds per minute you can shoot?  I have been letting mine cool about 5 minutes every 50 rounds in rapid fire, but I don't know if it is required or not.  

Your fine doing what your doing.  My brother shot his 90 round drum and then a couple of 30 rd mags in a row once... the barrel turned gray.  My guess (I'm a mechanical engineer) is that the molecular properties of the barrel changed... probably not a good thing.  I always know my AK47s are getting too hot when the plastic foregrips start melting and smoke big time telling me its time to take a break.

the gas tube will die well before the barrel does.

It depends on weather it is chrome lined or not. I have ran seven thirty round mags back to back without a problem and accuracy not effected. However as the barrel heats up it tends to warp a bit- you may notice some odd stray rounds the hotter the barrel gets.
AK's can go a hell of alot longer on rapid fire simply because the barrels are hammer forged making the metal quite a bit more dense than any AR barrel. Most AR barrels are rated to about 10K-14K rounds before needing changed. AK barrels are rated to 45K or better.
Really you need to decide if the rifle is a bench gun or a fun gun. I opted for the second which means burning it up. I have seen AR barrels glow from so much heat but once cooled accuracy tends to return to normal. You may consider Cryo-Teching the barrel for wear. This makes the metal more uniform which adds heat dissipating qualities= more accuracy. The barrels only turn "gray/white" because the park has been heated to much.  

Generally, a barrel turning grey or white isn't any big deal. All that has happened is you've heated the barrel to the point that you smoked off all the protective oil trapped in the parkerizing.

Wipe the barrel down with a little CLP and it will change right back to the original color.

I "white-barreled" a 16" AR last weekend and while that type of shooting probably isn't going to squeeze the maximum number of rounds out of my barrel life, it hasn't done anything to affect the accuracy or safety of the weapon.

In the service, we use to use a rule of thumb "literaly" if you could feel the heat from about six inches slow down a little. LOL

Actually advise you look at your manual. It will tell you how many rounds how fast.  Typically there will be a safety factor of about thee times built into it.