Any suggestions for the second problem?

Mathematica can handle the integral for specific integer values of the first parameter n, but not for general n.

Integrate[
 x^p Exp[-x^2/2] HermiteH[3, x], {x, -Infinity, Infinity}, 
 Assumptions -> {p >= 0}]

(* Out[246]= -2^(2 + p/2) (-1 + (-1)^p) (1 + 2 p) Gamma[1 + p/2] *)

This has to do with the interplay of how convergence testing is done, and how assumptions are handled-- in effect integrality is replaced by realness so as to avoid nongeneric oversimplifications in case of discrete-valued parameters. For general real n>=0 the convergence test fails at infinity.

Even if the convergence were correctly assessed, I'm not sure a symbolic result would be produced. Below I use GenerateConditions in a way that, I think, bypasses the convergence assessment. It comes back unevaluated. Integrate[ x^p Exp[-x^2/2] HermiteH[n, x], {x, -Infinity, Infinity}, Assumptions -> {p >= 0, n >= 0}, GenerateConditions -> False]

Out[247]= Integrate[ E^(-(x^2/2)) x^p HermiteH[n, x], {x, -[Infinity], [Infinity]}, Assumptions -> {p >= 0, n >= 0}, GenerateConditions -> False] Integrate[ x^p Exp[-x^2/2] HermiteH[n, x], {x, -Infinity, Infinity}, Assumptions -> {p >= 0, n >= 0}, GenerateConditions -> False]

Out[247]= Integrate[
 E^(-(x^2/2)) x^p HermiteH[n, x], {x, -\[Infinity], \[Infinity]}, 
 Assumptions -> {p >= 0, n >= 0}, GenerateConditions -> False]

I'm not interested in calculating the quantum mechanical expectation value, which is the moment of the square of the wave function. I'm using the QM Harmonic oscillator wave functions as a basis set in which to expand a function I want to compute the moments of, so I want the moments of the HO wave functions, not its square.

In my Mathematica Integrate command I included the restriction that n and p both be positive integers.

Since HermiteH[n,x] is a polynomial of degree n, the coefficient of Exp[-x^2/2] in the integrand is a polynomial of degree n+p. For large x, this behaves as x^(n+p) Exp[-x^2/2], which clearly goes to 0 faster than 1/x, so the integral converges..