I see you answered your first question yourself -the answer to this one is indeed what you are guessing: simultaneity. Put simply, if the box is moving, and the observer inside of the box measures the two light bulbs turning on at exactly the same time, then an external observer who is at rest will notice the light bulbs turning on at different times, assuming said external observer could see inside the box (which kind of defeats the purpose of having the box in the first place, of course). What the "at rest" observer will see is that the signal sent through the nervous system of the person inside of the moving box will be delayed, or accelerated, in such a way that the light bulbs will turn on at different times so that their light can strike the moving observer's eyes simultaneously.

Basically, the underlying principle behind special relativity is that there is no "at rest". If the walls of your box are transparent, then you won't be able to tell if the box is moving and the person outside is standing still, or if the entire world is moving by and the box is the only thing that is not moving. "At rest" is merely a convention that we used to describe physics relative to a very large object with a great deal of inertia - if you are at rest, and then you are not moving relative to the portion of the Earth that you are standing on. Of course, you are not at rest relative to the center of the Earth (unless you're standing on the pole), since the Earth is rotating, and you're not at rest relative to the sun, or the center of the galaxy, etc.

The main flaw with your conclusion lies in the basic tenet of relativity:
namely, the speed of light is constant in ANY inertial reference frame.Things will proceed as you describe them, but only for people in a frame of reference outside the box looking in. For the person inside the box, the speed of light is constant in the box's frame of reference. So events will proceed exactly as they would in a box at rest...because the box IS at rest in this frame (by definition)!

Relativity thus predicts some weird things, like the loss of simultaneity (e.g., the lights in your question turn on at the same time in the box's reference frame, but at different times in the outside reference frame) and time dilation (like your light clocks would show). But these things have been measured and shown to be spot on.

So, if you are okay with the speed of light being a constant in any inertial reference frame, and okay with all inertial reference frames being equal, then one has to be okay with the loss of simultaneity and all the other weird (and cool) things special relativity predicts!!