transfer the weight of the components to the top skin. think of a tray that the electronics sit in, bonded to the underside of the top skin. you can bond it to the bottom skin too, if you want, but the important loading is to the top.
the inturrupted rib isn't helping much. a lighter rib right on either side of the component tray will probably be better.
screws into the edge of MDF are pretty much useless.
don't notch the long ribs (the ones that turn the corner and bear the weight of the top) very much. don't make them from MDF either. the transverse ribs can float between the long ones. it will be convenient for assembly to fasten them into a frame before applying the skin, but not necessary for the strength of the finished article. try to either limit yourself to fasteners that don't displace a lot of wood (like finish nails) or have correct pilot holes for the screws you use.
shifting the notches (staggering the transverse ribs) is fine, and will help avoid weak spots lining up with each other. better is to not notch.
this assembly is similar to a hollow core door. in a door, the function of the ribs is taken up by some lightweight, more-or-less continuous material like foam or cardboard. you need to bear weight against the panel, so you will need a thicker top skin and some ribs running the long way. other than that, though, it can be a lot like a hollow core door. the cross ribs can be replaced with shaped blocks of foam, if you want. the bottom skin, the harder one to assemble, can be thinner that the top. if you can, the cool way to make the ribs is to either steam bend or laminate hardwood.