Thoughts on problem 3:

def P1(): sumU = 0; for(#U=1; #U<3^^^3; #U++): if(#U encodes a well-defined boundedly-recursive parameterless function, that calls an undefined single-parameter function "A" with #U as a parameter): sumU += eval(#U + #A) return sumU def P2():
sumU = 0; for(#U=1; #U<3^^^3; #U++): if(#U encodes a well-defined boundedly-recursive parameterless function that calls an undefined single-parameter function "A" with #U as a parameter): code = A(#P2)
sumU += eval(#U + code) return sumU def A(#U): Enumerate proofs by length L = 1 ... INF if found any proof of the form "A()==a implies eval(#U + #A)==u, and A()!=a implies eval(#U + #A)<=u" break; Enumerate proofs by length up to L+1 (or more) if found a proof that A()!=x return x return a

Although A(#P2) won't return #A, I think eval(A(#P2)(#P2)) will return A(#P2), which will therefore be the answer to the reflection problem.