That would work, but it s not actually how I did those particular ones.
I did those on the table saw as a "cope and stick" construction using a dado blade (a stack of two ordinary blades and a pass from each side would also work well). That put the panel in the middle of the door, set back from both sides:
Then I cut the rails to be final length + 2 x Depth of slot, and cut a full width tenon on the end (few passes over the blade with the cross cut guide in the table slot) - just the right size to fit the slot.
The whole door could then be assembled. Since the panel was MDF and all the lengths were long grain, the whole thing could be glued.
If going for a real wood panel, then I would cut that a little under full width, and leave it floating - with just the rails and stiles glued[1]. Looking at the top of the door you can get a glimpse of the construction, like on this real wood version:
The really simple way is as you say a panel the size of the door, and then "planted on" rails and stiles. All from MDF.
The youtube master of the MDF door must be Peter Millard. He has loads of videos on these done using a number of techniques. e.g.
Well the fronts can be any panel assembly method you like - including planted on rails and stiles.
The boxes are in most cases not that critical. I used finger joints since I can do lots of drawers in a single operation, and they are very strong) 90 odd CDs are quite heavy! Having said that, there are plenty of easy ways that work just fine. Simple gluing and screwing but joints at the 4 corners of the boxes will be fine. You can then plant on a base, glued and pinned, or better still just route a rebate round the whole of the bottom of the drawer box, and drop the draw base into it - that looks better and adds more resistance to "racking" in the base.
Another option that works well are the drawer mitre lock router bits:
It was "ok" in the form you see there. In reality I went back and changed it a bit (must do so extra photos for it). Later I fitted a much heavier AV amp (15kg), and found I needed to add a third drawer runner under the middle of the base. I also realised that when seated you could not actually see the top of the box much beyond the leading edge. I was also always a bit annoyed by the slight variation in margin around the edges caused by an earlier "out of square" cutting error. So I modified the box to fix the "square issue" and also remove the entire top of the thing apart from a 6" deep strip at the front. That made access for wiring up *much* better. I also made a slotted vertical divider to go in the front - that allowed me better control of the margin spacing, and looked a bit more appealing.
In which case you already have more than enough kit to go about these in a number of ways :-)
Yup, sorry my bad - softwood!
Most DIY shops have a stock of all kinds of beads and profiles including dowels, coves, quarter rounds etc, but also usually just plain old strips of something rectangular you can trim to length and stick on.
However given the tools you have at your disposal, simply ripping strips of the side of a suitable thickness of plank will give you plenty of shelf or door edge lipping material.