Can I dye my carpet? The color look so different in the store...

Only if it is made from a natural fiber, like wool, silk, or jute. However, the vast majority of carpeting is made of oleofin, tradename herculon, or some blend of polyester. These man made fibers are colored using very high heat, hundreds of degrees, which is why the color does not "run" or fade like dyed natural fibers tend to do.

i wouldn't. But you can paint an interesting design, stencil, or border.

What about adding a darker accent rug over it?

I wouldn't risk it. Place a pretty accent rug or if you are really wanting to tackle the project I would test a small area of the carpet maybe on a corner or closet to see how it really reacts

lighting also affects the color appearance...different florescent bulbs (which is typically used in stores) create different colors....cool white, soft white, daylight will each produce a different shade...I suggest a cheaper idea is to use sunblocking curtains. This will automatically darken the appearance of the carpet...in the future, for true color appearance ask to take the sample outside the store and see it in full sun light