I'm having issues with the Fate Point system. Does someone spend a Fate Point to "discover" an aspect?
Yes, you can spend a Fate point to "discover" an aspect. That's called Declaring a Story Detail.
That's not the only way to discover an aspect, though: you can also create an advantage, or use a stunt that creates aspects. One of the examples for Creating an Advantage with Crafts is "discovering" a vulnerability in a crafted work.
Now, from what I gather he can still attack- because he didn't do the "Create Advantage" action, instead he discovered an aspect that's logical on a scene with "Dark and Stormy" on it. Now does he spend a FP to invoke that aspect?
There answer here depends on the method he used to create the aspect.
He used a fate point to declare a story detail and get that aspect for free. That doesn't come with any free invocations, so he'll have to spend another fate point to invoke that aspect.
You might find that pretty hefty, but declaring a story detail is pretty powerful. You can create aspects your own stunts or skills don't allow you to create, and aspects are always true, which means the aspect can immediately begin to have mechanical and narrative impact. You just don't get a free invocation along with all of that: that would be effectively giving you a refund on your fate point, and make all that other stuff just happen for free.
If he had taken a turn to use Create an Advantage and succeeded, however, he would have received a free invocation on that aspect, or two if he succeeded with style. Then in his next turn, he could use a free invocation as part of his sneaky attack.