Consider the subset U ⊂ R3[x] defined as

[Karl Porter]

The part above is OK.
No. Take a closer look at post #56. I've laid it all out for you, including how to get a set of basis functions.
I have no idea what you're trying to say here.

Basis would be {x^3+x^2,x^3-x}

 

[Mark44]

Basis would be {x^3+x^2,x^3-x}

Yes.
Let's call these $p_1(x)$ and $p_2(x)$, respectively.
As a check that these functions are in U, it would be good to verify that $p_1(0) = 0, p_1(-1) = 0$ and that $p_2(0) = 0, p_2(-1) = 0$.