Plumbing – Drain pipe too high for garbage disposal

The drain into the wall needs to be below that of the disposal and sink, there's no way to beat gravity with this. Your options are to:

1. Raise the disposal output (with a different disposal)
2. Raise the sink (with a different, shallower sink, or at least a shallow drain for the disposal)
3. Lower the drain at the wall

Your math looks good, but you haven't accounted for the tail-piece from the disposal into the trap. The 7 3/16" you calculated is to the center line of the disposal's outlet, and there's a piece attached that fits into the trap. In the linked diagram, that tail piece is shown as 4" high, so in the worst case, measuring from the bottom of the countertop down, you'd need 9" for the sink + 7 3/16" sink to outlet + 4" tailpiece = 20 3/16". So that wouldn't work. But on the shallow bowl of the sink, you're just under 18" and should be OK.

You do have some additional wiggle room in two possible areas:

1. The tailpiece doesn't need to fully extend down. Your trap can come right up to the part where it curves to join the disposal. This can probably reduce its height from 4" to 3" or maybe even 1.5 to 2".

2. Hard to tell from the diagram, but the height of the disposal plus drain flange should be measured from the top of the flange not the underside of your sink. The flange drops into your sink from above and is clamped on from below, so whether your sink's made of very thin material or very thick stuff only affects the position of the clamp, not the height of the final assembly. (Unless you use an extra-long flange, which might be necessary in e.g. a thick stone sink.)

You will also need a bit of extra height to allow for proper sloping of the drain lines. 1/4" or 1/2" could be fine. I'm also assuming your 18" is measured to the center of the wall drain.

So in short, I think you will be fine if you're willing to put the disposal on the shallower end of the sink, and you have a chance of making it work on the deeper end too. Good luck!