$\operatorname{span}(x^0, x^1, x^2,\cdots)$ and the vector space of all real valued continuous functions on $\Bbb R$
Solution 1:
Exactly.
4. doesn't hold in general, so neither does 1.
2. and 3. are true, $\mathcal P$ is a subspace of the space of all continuous functions, and the polynomials $p_n$ are linearly independent from each other.