Hanging a shelf on a difficult wall

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.

Disagree, aquariums do not hold most of their weight under the corners. The weight is almost uniformly distributed on portions that are in contact with the supporting surface. If it is an acrylic tank with no frame, then all that weight is distributed across the whole bottom surface (assuming it is resting on a shelf like you illustrate). If it is a glass tank with a frame, then the weight is around the perimeter wherever the frame contacts the surface.

The rod under the corners are good, and your idea about diagonal members will prevent sagging. But that is probably overkill and also adds complexity to the fabrication. I would just run lengths of angle iron between the rods along the long length. I don't think sag would be an issue on the short length.

You didn't ask, but spend the extra money and get an acrylic tank (not just any plastic tank though, acrylic). A glass tank will eventually leak especially if it is not on a perfectly flat surface (flat, not necessarily level), and sometimes even then.

Lastly, that entertainment center is probably made of MDF or a similar material. It will swell like mad if left in contact with water long enough. No matter what you do, you will end up with some water outside the tank at some point. So, this is not the best material overall so do take care.