The H2S theory of ME/CFS was based on the fact that if you give mice H2S, they go into hibernation, ie, a state of very low energy usage, reminiscent of ME/CFS. It was theorized by Marian Dix Lemle that the same thing might occur in humans, that H2S might trigger a hibernation state, and that ME/CFS might thus be caused by overproduction of H2S in the gut, creating human hibernation, which we diagnose as ME/CFS.
KDM picked up on Marian's theory, and started offering his "neurotoxic metabolite test" for H2S. But humans do not react the same way to H2S as mice do, and I don't think there is any evidence that H2S can cause a hibernation state in humans. So although Marian continued to pursue her theory, I have not heard anything more about it
I don't know about the affect of H2S on the gut, but the paper you linked to looks interesting. It seems that high H2S is linked to impairment of detoxification in the gut mucous membranes.
Small intestine bacterial overgrowth (SIBO) can result in production of H2S, as well as hydrogen gas and methane gas. The last two are measurable by the hydrogen/methane breath test for SIBO. Last time I checked, no SIBO tests were available to detect H2S, but this was being worked on. So it's possible you might find a SIBO test now which checks all three gases.