How should this microwave shelf be stabilized

Yes. They can easily take the weight when anchored to studs. I use it for my computer desk. It works great and gives you a very inexpensive, height adjustable, and narrow depth desk. I have two shelves. One is 24" deep and is a keyboard/work surface. The other is just 12" deep and holds my LCD monitor.

I opted to place just two rails ~32" apart (skipping one stud) so that my knees don't bang the brackets. This causes the desktop to bow very slightly under the weight of my hands. I suppose this could be fixed by using a more rigid surface vs. the default melamine shelving.

My PC is on the floor but these shelves could easily hold the weight.

A 2 piece shelf is a good plan, but my scheme differs from yours from there. Yours would work, but mine's better (IMO). Instead of plywood, I would use MDF. It's dimensionally stable, easy to work, and paints up very nicely with a very smooth surface.

I would cut a circular hole to fit the duct closely so small things don't fall into the void below. Consider buying a small electric jig saw for sawing. It doesn't have to be very expensive, you will likely find other uses for it once you have one. And it'll save a ton of manual labor! Sell it on flea-bay when you're done if you really don't want it. This all can be done manually if you're a glutton for punishment.

You don't need angle brackets, though they could work. Gluing and screwing MDF cleats to the walls will be much stronger and probably more accurately assembled. You don't really need to attach to the cabinet sides at all. A cleat along the back wall and a front filler panel would be adequate support. If it's not too much trouble, the extra support would not hurt though. I would use construction adhesive for glue.

You will need to pre-drill all fastener holes in MDF. It's too dense to drive anything through it without doing so. You should not use glue to hold the actual shelves, screws alone will allow removal to access the range hood or retrieve items that fell through near the duct. You normally should never run fasteners into the edge of MDF, but in this case it's non-structural just to keep the shelves from moving. Be sure to pre-drill the full screw depth into the cleats because fasteners in the edge of MDF cause it to spall very easily if not pre-drilled.