I want to go for a natural, light look so can I use maple wood without
a stain?
Yes, you can use any wood you want without stain, there is no rule that says you need to stain wood. The finish you choose can effect the color to some degree so testing on some scrap is a good idea. Every type of wood has unique characteristics, i can not list them all here.
Are the two side pieces necessary?
The breadboards do serve a function, while you could technically get away with not using them ( because of the pocket hole joinery of the other boards ) they tie all of the ends of the other boards together. If you did not tie them together then each board could/would twist and/or warp, resulting in the end of your table being uneven and wonky.
That being said, breadboards done with pocket hole joinery are not really accomplishing the structural integrity that breadboards are designed to provide.
What type of wood should I use? Home depot calls the 2x4 Maple pieces
"studs", which sounds more like structural wood. Will that be okay for
a desk?
You can choose any kind of wood you like based on your own criteria I.E. color, grain, hardness, stability etc. Make that decision base on your own research. Maple is an excellent choice, it is used for butcher block worktops.
It does need to be thick enough to be stable for the dimensions you choose. You may want to go to a dedicated lumber supplier instead of a home improvement center where the selection is limited. ( i would be shocked if the Home dumpo has 2x4 Maple )