Personally, I like the hottest iron i can find, im not very patient, and sometimes i burn things, but Im the person that will have to fix it, so its not an issue for me. I think around 350c is my starting range, when i use temp controlled irons, I usually set them between 750-850F, usually in the 850 range. You probably dont want to go that high, because you really have to move quickly to avoid burning things.
You may have an ESD or EOS issue going on. I have had some strange grounding issues with my non-grounded soldering iron, It actually lit up LEDs(just a tiny bit) while I was soldering them. It maybe that you have something like that going on.
All LEDs are not the same, so it maybe that the ones you got are particularly sensitive. I would do some testing on the LEDs and see if you can cause failures, and try to figure out what not to do, before you ruin all your LEDs!
I tried to load up that page you linked, and it didnt load for me.
Make a new grid, and this time pre-tin the wire, make sure the solder is adhearing nicely to the wire, but just leave a little film of solder on the wire, not a glob. then when you solder your grid together, just reflow the solder. Flux is important because you want the solder to flow quickly, so you can get the heat off as fast as possible (1-2 seconds). The flux also helps prevent "Ice sickles" so you dont accidentally short on anything. I have reflowed the solder many many times without destroying the part, so I know that they can take a lot of heat, but sometimes it may wreck them with only a few seconds of heat.
How are you testing your LEDs, maybe that is causing extra stress?
I dont know how your LEDs could light or not light when the should/shouldnt, but my only guess is a short/open situation. I think its possible for an LED to fuse, and create a short, but my guess is its usually an open.
I reloaded the page, and now it loads, so i'll look it over.