You would definitely need a router table in order to route long edges without any flaws!
But as this photo from
@MrBoat 's shop shows, there is no reason for any woodworker not to have a router table, they are far too useful and far too easy to make!
(I would probably go to Lowes or HD for one of their"scrap" pieces of linoleum covered counter tops, I suspect MrBoat threw this together from scraps he had on hand).
As for point two, these joints fit without having to force them. You would never use a rubber mallet on them! I would route both edges of all four pieces - both sides, the top, and bottom. Use a chip brush or air to make sure no saw dust remained on the surfaces. Apply glue to all surfaces, and use a band clamp to squeeze out the glue and pull joints tight, "wiggling" the box every now and then as I tightened the clamp to help insure all surfaces are finding their new homes. None of the routed profile corners are less than 90 degrees so they are not particularly frail, but there is no source of resistance to require forcing it together.
You may experience some resistance using real wood because the panels being joined may be warped and you have to pull the warp out of the wood as you clamp them, but not a problem with mdf panels.