May have a solution to the 2nd question now. Seems kind of backward though...

Let x be in V.

Assume {u+v+w, v+w, w} is a basis for V.

Then x = a(u+v+w) + b(v+w) + cw

Rearranging yields:

(*) x = au + (a+b)v + (a+b+c)w

All of the coefficients above are in F by properties of F.

So then (*) is true by hypothesis, since that is the definition of {u, v, w} being a basis. QED.

Does that work?