You can glue a matching size piece of MDF onto the bottom, which will help keep it flat. Get your plate first. I prefer aluminum, and with a twist lock insert so it's quick and easy to change bits. Once you have that, you will want to transfer the outline onto the laminate and cut that opening so it's a fairly snug fit, but not too tight (a PITA for removing the router). Then glue on the MDF or make a truss as suggested to help keep the laminate flat. If you use MDF, position it, draw an outline about a half inch smaller than the opening, then cut it out to create a "shelf", then glue the MDF on. I would not use contact cement for this, too easy to mess it up.
You need to set up some sort of leveling adjustment. You could do this with screws with the tips ground flat, but if you are going to put any screw in MDF, you have to pre drill. You could put in Tnuts and put in bolts with a lock not to hold the height, but it's far easier to buy a set of 4 leveling gadgets (pix) and mount them in the four corners.
Personally, I'd use the MDF rather than truss because I think it will be easier to mount on a table than a truss system. My TS outfeed table has a truss and it has stayed flat for more than a decade.
I like the idea of an enclosed cabinet, or one with drawers to the side of the opening for the router, and also below that opening.
Stick some time ago suggested leaving room for a "snorkle" that pulls fresh air up through the bottom of the router for better cooling and to reduce sawdust incursion into the router. I haven't done this because my table would have to be remade. But I really like the idea.
You can put a hinged door over the router opening, but I'd think seriously about making it with a glass or plactic window so you know when a cleanout is due.
Dust extraction under the table is usually done with a Y splitter behind the table with a larger opening through the back into the router chamber. You must also have a way to let air into that chamber or there will be no air flow to carry the sawdust away. I'd bring air into the chamber through the same channel the snorkle runs through. You could also place a filter over that snorkle opening if you are a fuss budget about dust collection.
Your fence should also have a 2.5 inch dust collection port attached just behind the bit. That will help pull sawdust up and away from your router. I know Rockler makes a little port like this and I'm sure other makers have them too. My experience is that you want ALL of your DC connectors and hoses to be one brand. 4 inches isn't 4 inches from manufacturer to another. Parts are NOT interchangeable. I only have Rockler store access.
The diagram shows the layering.
The more I do, the less I accomplish.