Learn English – What’s the generic word for weekly/monthly etc. service

Yes.

It’s very tempting to leave the answer at that, but… some more notes.

1. The most cursory look at any newspaper, or around the blogosphere, confirms this answer. ‘Last week’ is common as muck, while ‘yester-week’ is virtually non-existent, and when it does appear, it’s for archaic effect.

2. ‘Last week’ isn’t in any way incorrect. It’s had the meaning ‘most recent, latest’ for over 600 years: in 1411, for instance, the rolls of Parliament under Henry IV refer to “The last parlement…” (OED.)

3. As your sources point out, however, ‘last’ can sometimes be ambiguous, and ‘latest’ is often a good alternative in those cases. Talking about ‘the latest week’ sounds odd and affected, just as it did when Fowler wrote; but talking about ‘Madonna’s latest single’ is perfectly fine.

4. ‘Yester-’ isn’t actually as archaic as we tend to think! A few examples — ‘yesterday’, ‘yesternight’, ‘yester-eve’ — are old, but (according to the OED again) the general combining form (in eg ‘yester-week’) only really shows up in the 19th century. Amazingly, even good old ‘yester-year’ was apparently only coined in 1870, by Rosetti translating Villon: “Where are the snows of yester-year?”

5. Finally, none of the sources you give support your implication that ‘yester-week’ is the correct usage that should be preferred to ‘last week’! The Bierce and Fowler quotes point out the illogic and occasional ambiguity of ‘last’ (note that neither of these makes it grammatically wrong); Fowler’s suggests ‘latest’ as a sometimes better alternative, but neither even mentions ‘yester-’! The OED simply points out that ‘yester-’ exists (at least, existed for a few decades) and means roughly the same thing.

Participant might work. I'd leave off the "service" adjective if you use your terms mostly in contexts where it's understood.