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.