Never make the mistake of letting people know you can sharpen knives. Apparently it's a lost art, you'll end up with a tray of them in your lap when you're over for movies, in my experience.
Yes, it would be. The main reason things aren't dishwasher safe is the detergent or heat can damage the material. Food grade stainless steel would be fine with both.
Just so you know putting blades through the dishwasher dulls them faster, the main thing is the edge will warp due to differences in temperature so you'll need to hone them with a steel afterwards to keep the edge consistent, you'll also probably need to use a whetstone more often.
But up to you, I've got 'trash and bash' knives I put through all the time, but I keep my expensive knives away from it
And, how can you tell it was sharpened from the picture? Usually, when I've seen sharpened knives, the cutting edge has scratch marks perpendicular to the blade and is wider than normal from being grinded into a longer slant.
Friendly reminder that most home kitchen knives don't need frequent sharpening. Honing the blade is usually all that's necessary unless there's damage to the edge (which shouldn't happen much if you keep your blade honed).
I haven't sharpened my daily driver chef's knife in 7 years, and it can still slice through a tomato with no effort.