Anchor 1×2 wood on cabinet side wall

This is a tall unit and if it were fully loaded, it could really hurt someone (especially a child) should it fall. Products like the 3M command strips are designed for hooks where the force on them is acting downwards. Your cabinet on the other hand, should it tip, would be pulling away from the wall and I really don't think those strips would hold. If it were my house and my kids, I would not trust them and would opt for a bracket.

As for glueing it, in all liklihood, it would just pull the paper layer of your drywall off with it. Even if it held, it would likely damage your walls more than a couple of holes for a bracket when it comes time to remove it.

For a solution to this, what I often see come with tall bookcases is a short strap (webbbing) with a screw and washer on both sides. One side screws into the back of the bookcase, and the other should be screwed into a drywall anchor - ideally a toggle bolt since they don't easily pull out. The washer is important to prevent the screw from pulling through the webbing. This particular one below uses velcro, but I'd personally screw it into the back of the bookcase.

_{(source: themanlyhousekeeper.com)}

Alternatively a L bracket will work, but they tend to look ugly and depending how much clearance you have, can be really tight to install and/or remove.

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.