Electrical – Dryer Outlet in the kitchen

Replace the outlets with GFCIs -- the operation of a GFCI in no way depends on the presence of the equipment grounding wire. You'll have to use the "No Equipment Ground" sticker that comes with them, by the way.

P.S. on the metal boxes -- since it sounds like your house is a mix of K&T and other wiring techniques (perhaps NM additions), you cannot ground to the boxes -- that technique only works if your house was wired with BX/AC, MC, or metallic conduit (RMC/IMC/EMT).

(Confirming that you have K&T and NM mixed in your house, based on your photos, by the way -- I also do not see a ground wire on the NM, but the angle of the photo makes it hard to tell.)

I recently had a problem that stumped me. My dryer had worked for a couple of years and then stopped heating. Turns out, the 240 receptacle had a bad contact on one of the 120 legs so it made poor contact with the plug. I replaced the receptacle. Later, I autopsied the old receptacle, cutting it open by drilling out the central rivet, and found that one of the spring clips had actually broken, so it didn't maintain tension. Of course when I had probed the receptacle I had been able to get the 120 because I was hunting for it. Perhaps your receptacle is bad?