What tool or program would you recommend for learning touch typing on Linux?

I don’t have a particular recommendation, but in addition to local applications, there are also websites that you can try. The keyword you want to search for is “typing tutor”, e.g. on Debian.

For someone whose native language isn’t English, would you recommend learning on their native keyboard layout or switching to the US QWERTY layout for programming purposes?

For programming, the most important thing is that you can type the full set of characters present on US-QWERTY without too much acrobatics, because programming languages tend to use all/most of them.

Other than that it’s just down to your preference and comfort. I don’t recommend putting stock on any hype related to typing speed.

I’ve tried two different setups for Hangul input.

The first combination (GNOME+X11+IBus) just worked—it was trivial to setup from the GNOME Control Center and there were no issues that I could find. Last time I tried this was around a year ago but it had been working for years before that.

The second combination (Plasma+Wayland+IBus) barely worked: I could enter characters but couldn’t add space or other symbols between characters, and settings were all over the place and would randomly stop working. Last tested ten minutes ago.

I’ve heard people getting better results on Plasma+Wayland+Fcitx5 but it’s not something I’ve tried.