Is it valid to have a <section> tag inside another <section> tag?

Yes! w3 encourages you to explicitly wrap sections in elements of sectioning content, instead of relying on the implicit sections generated by having multiple headings in one element of sectioning content and nested <section>'s are allowed.

Yes, since a website is generally divided into one or more sections, use it as long as you feel it describes the structure. A layout can have 1 or even tens of SECTION elements, just know that it is NOT a DIV replacement :) DIVs are still used and recommended (usually for grouping means).

Yes, that will work fine. At least, it works for me.

You can try it out here.