Here is solution to #2. By Weierstrass Approximation Theorem there is a sequence of polynomials p_n(x) converging uniformly to f(x). But \int_0^1 p_n(x)f(x) dx = 0 (because if you write p_n(x) = a_nx^n+...+a_1x+a_0 it should be clear by hypothesis). But then \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.