Now the real question is: what is the problem with the iron? And is it just unsafe for the LEDs or is it also unsafe for me to touch? How can I get around this problem? Should I take it back to the store saying that it's not working properly? Or is this something normal? Assuming that this should not happen normally and that it's not just a problem with my iron, but a problem with all the irons of the same type, what kind of soldering iron should I get instead? What detail should I pay attention to?
Why don't you take a AC voltage reading between your iron tip and an independent path to your power ground when the iron is turned on and see if there is indeed some AC leakage voltage on it. Most soldering station irons have a positive direct ground connection to their body/tip as a protection against ESD damage, but yours sounds like it might have a defect or broken connection internally or something like that.
Most LEDs have a maximum safe reverse voltage rating of only 5vdc or so, and it doesn't take much current to damage a led that way, just apply reverse voltage above it's safe rating.
Lefty