Here is solution to #2. By Weierstrass Approximation Theorem there is a sequence of polynomials $\displaystyle p_n(x)$ converging uniformly to $\displaystyle f(x)$. But $\displaystyle \int_0^1 p_n(x)f(x) dx = 0$ (because if you write $\displaystyle p_n(x) = a_nx^n+...+a_1x+a_0$ it should be clear by hypothesis). But then $\displaystyle \lim \int_0^1 p_n(x)f(x) dx = \int_0^1 f^2(x) dx = 0 \implies f(x)=0$ (this is were we use uniform convergence). Q.E.D.