You've got about 93 issues. Let's try and take a few and make some sense of them.
The super cheap Henckels are very hard to sharpen under the best of circumstances. Your carbide sharpener doesn't actually sharpen in the sense of creating or maintaining a fine edge. Rather it rips a bunch of metal off the knife and leaves a very coarse edge which will dull quickly. Your intuition that it's a good idea to up the knife quality and the sharpening quality is good.
If you can sharpen carpentry tools on bench stones you've got a pretty big head start compared to people who've never sharpened at all. If you understand the process of raising a burr and deburring, and can accomplish them, you've got a huge head start.
"Clicking in" is useful with kitchen knives, but nearly as much as it is with carpentry tools nor is it as easy because: (1) Kitchen knife bevels are more narrow than tool bevels; and (2) the natural tendency of the edge angle to become more obtuse if the bevel shoulders are not well defined (partly a function of their width), as the knife is repeatedly sharpened.
But learning to hold an angle, thin, and all that other knife specific jazz isn't too much of a stretch. You'll have to work at it, but not that hard.
Both gags IceMan recommended work pretty well, but not as well as good bench stones.
Let's hold of on comparing costs between gags and stones, and between the various stones, until you decide what kind of knives you're going to buy.
Yes to drywall screen. Of all the inexpensive ways to flatten it's the fastest and best -- and I think it's your best current option. However, it's not particularly fast or good and if you get serious you'll want better. Unfortunately "better" enough to be "better" is something like a DMT XXC for $80ish.
If that's what your wife wants, get her one. If you want a 10" chef's/gyuto, get one for yourself. Will your wife fall in love with your knife? Maybe.
You have a range of options. The minimum amount of money you can spend and still get a quality knife is probably a Forschner Fibrox or Rosewood. Considering how you've described your wants and expectations the big downside with a Forschner or any knife made from a similar alloy is not sharpness so much as maintenance. They need frequent (and competent) steeling.
It sounds like your really asking about something the next step up and Japanese. Fujiwara FKM, Kagayaki VG-10, MAC Superior, and Tojiro DP santokus are all within your price range and all at a fairly similar level of quality.
Without getting too deeply into the specific plusses and minuses of VG-10 versus anything else, the Kagayaki is "mono-steel"
VG-10, while the Tojiro DP is a triple layer laminate (san-mai) with a VG-10 core (hagane) to do the cutting clad between softer outer layers (jigane).
They're all pretty good, but none of them great.
I hope this helps, and please feel free to ask more questions,