Hey @maknib, welcome back. 
Pretty much yes these days. (There are ways to account for older browsers, if you need to.)
Certainly not should. Well, they all do, really, because you can view any site on mobile. But there are CSS ways to make the layout better on small screens—via CSS media queries—that have become popular.
Combo of both, depending on the need.
Also optional. WP is still popular, but there are many great new options out there, such as Statamic and Craft, to name but two.
Better to find a project that actually needs to be built—even just something personal.
Remember the Nike motto. 
If you need motivation or inspiration, consider one of the great courses offered at SitePoint. E.g. https://www.sitepoint.com/premium/courses/getting-started-with-css-2903