Honestly, I wouldn't try to solder my own BGA's. I know this doesn't directly address your issues, but hear me out.
It takes a lot of work and effort to solder a BGA. There's a lot of trial and error. A lot of messed up test boards. But then it's soldered. Now what?
Now you have to prove that it's soldered correctly. For that you need one or more of the following: JTAG test (US$10k, never has 100% coverage), optical inspection (US$20k for the equipment), or X-Rays (US$500k). The cost of doing these tests is too much for the normal hobbyist, and is even beyond many small companies.
Skipping those tests, you proceed with debugging your PCB. And let's say that the BGA is a complex CPU. Inevitably you'll find a bug. The CPU will randomly crash. Is it your software, your electrical design, or the soldering on the BGA that's causing the problem? Debugging this, in light of some possibly problematic soldering, is going to be terrible. It will add a lot of time to your debug process, possibly months, and you'll loose a lot of hair on your head. And then you can repeat this for the next major bug.
Without confidence that your soldering is perfect, you will always have this dark cloud over your head. Every little bug that shows up "could be a BGA soldering problem". This is made worse if you have multiple engineers working on the same PCB since the software guy will be questioning the hardware guy, etc.
Then, even if the BGA soldering is perfect, did the chip get too hot? Did you destroy the chip by getting it too hot? Even on modern assembly lines this is an issue. But with the proper equipment you can adjust and measure the temperature profiles to at least get you in the right ballpark. On one board I did recently, the BGA's were being damaged. The solder balls looked great, but under a very nice X-Ray machine we could see that the gold bond wires melted from the heat.
I've been there. Not at the hobbiest level, but professionally as we were bringing up new boards while the assembly shop was learning to do BGA's. We had no JTag. No optical inspection. And the X-Rays were terrible. Our PCB had 11 BGA's on them. That was 2 years of hell I don't wish to repeat.
So, here's my recommendation:
Get someone who has the proper equipment, training, and experience to solder your BGA's. There are a lot of contract manufacturers that'll do a single BGA. It takes money, but that's way less than the time you'll spend trying to debug your own soldering.
If you must do it yourself, then you should get the proper equipment, training, and figure out how to get the experience required. For this to pay off in the end, you need to have a large enough company and need to justify the huge amount of time and money that you'll put into this.
But I would never try to just kludge something together. That's a recipe for, um, bad stuff.
Very good idea to practice, practice, and practice some more on salvaged boards until you get competent with new tools. You didn't notice the solder had reached the molten state because there's so little of it used to solder components.
The typical lead-free solder has a melting point around 217 degrees C, so you'll have to get the leads and pads up to that temperature before trying to remove the component. The reason why you need a much higher temperature is because you want to get the solder joints to the melting point as fast as possible. If the hot-air gun is set to a much lower temperature, it'll take a longer time to reach the melting point. The longer the time to raise the temperature of the leads/pads will increase the overall temperature of the component, possibly past the point of destruction. So the technique is to heat it up fast, remove the part and hot-air gun, then put the part where it can cool off.
Now, if you're removing a part that's already fried due to some other reason, no worries then. Just don't damage the board by overheating the pads and causing them to lift off. If that happens, your headaches are just beginning on this repair.
Best Answer
There's no way to be sure about potential damage without knowing the part number of your LED, which is needed to obtain its datasheet. The datasheet will outline the temperature specs of the LED (including max storage temp and max operating temp) as likely also its performance characteristics at various temperatures. I picked three random SMD LEDs from Digikey as a small sample (1 2 - the third is 160-1446-2-ND, but I lack the repu to post the link), and they have operating temps up to 80-85 degrees C. Max storage temp for one is 120C, but the other two max at 85C. Because of this variation, you need to know exactly what the specs of your LED are, but you'll might be cutting it a bit close. Some datasheets outline reflow info for their part, so your best method will be to follow that guide. Personally, I wouldn't be very worried about doing what you're doing as long as you have back up LEDs just in case, but if you don't have extras and/or can't risk damage to the PCB, then you'll do best to check the datasheet for all the info mentioned above.