Proposal to change default skin to Timeless
No, the message documentation windows (just like the user boxes mentioned below) have no such issues on mobile with Vector—since Vector is not responsible, mobile browsers use a so-called “virtual viewport” as a self-defense, which is much wider than the actual screen width (see Using the viewport meta tag to control layout on mobile browsers on MDN). This means that the text is maybe unreadably small, but at least the layout is not broken. Of course, this situation is much improved by the introduction of Timeless if Tux is made work well with narrow viewports.
For SEO: I see. This is a tough decision, however, as I still think that keeping consistency with Wikipedia is important, but I agree that letting developers find this site is just as important.
P.S. What about the site notice? Is there a reason not to increase its font size?
Regarding issues 1 and 2 (the navigation tabs), I see no easy way to fix it. I am not skilled enough to make distinct icons for them. I considered moving them elsewhere, but I have no control over where (filed Phab:T251258).
Regarding the 3rd issue, it would fall under Phab:T146134. Unfortunately I don't think I can fix it without re-doing the UI with a more responsive grid system, which is a lot of work.