(* Question 1 *)
dp=Function[{f,g},Integrate[f*g*Exp[-x^2],{x,-Infinity,Infinity}]]
nm=Function[f,Sqrt[dp[f,f]]]

N0=5
For[k=0,k<=N0,k++,
    w[k]=x^k;
    For[j=0,j<k,j++,
        w[k]=w[k]-dp[H[j],x^k]*H[j]];
    H[k]=Simplify[w[k]/nm[w[k]]];
    Print["H[",k,"]=",H[k]]]
(* The desired orthogonal polynomials are below *)