I'm having this problem too, and have for several months. There is a conflict with the WoW launcher and something in my computer. I click the launcher, it opens, and when I hit "play", it disappears entirely. I mean completely gone. It is nowhere to be found. I can't get into the game at all, nor can I close out the launcher since it disappears. I can't run the troubleshooter because it wants the game open and I have no way to open it. In order to get "out" of the launcher, I have to restart my computer every time, and then I get the dreaded blue screen.
I've been working around this by going to the battle.net site and actually downloading the whole game which takes an hour or more. When the game downloads, it doesn't require me to click on the launcher, it just finishes the download and I can hit "play" and it works. As long as I don't ever close out the game completely, I can continue to go back into the game from the gallery. However, once the game is exited completely, it is a problem again.
Today we had a brief power outage. I sighed as my computer re-booted, then went to the battle.net download site as usual, began the download, but halfway through the download, it stopped and I got the blue screen. This hasn't happened before. Obviously something is very wrong. I am not a techie, and I have an Alienware pc, only 2 yrs. old. My warranty is up, of course, and most computer shops here won't/can't work on Alienware. I finally found one who says he has the expertise for it, so will likely be taking my pc in very soon. It has been very disheartening trying to play, to say the least.
And by the way, I do use Firefox, but so does my husband, and he has no conflict whatsoever with his PC, an older Dell. I also have an Asus gaming laptop with Firefox, and there is no conflict there either.