Questions on $x^2+y^2+z^2$, $x^2+y^2+2z^2$ and $x^2+2y^2+3z^2$

Question

Question 1. My computation in 2018 suggests that a sum of two integer squares is a sum of three nonzero integer squares if and only if it is not of the form $$4^km\ \ (k=0,1,2,\ldots;\ \ m=1,2,5,10,13,25,37,58,85,130).$$ How to prove this?

I have two similar questions formulated in 2018.

Question 2. If a positive integer $n\not=1,5,29,65$ is a sum of two squares, how to prove that $n=x^2+y^2+2z^2$ for some integers $x,y,z$ with $z\not=0$?

Question 3. If a positive integer $n\not=1,17,2^{2k+1}\ (k=0,1,2,\ldots)$ has the form $x^2+2y^2$ with $x,y\in\mathbb Z$, can one show that $n=x^2+2y^2+3z^2$ for some integers $x,y,z$ with $z\not=0$? If a positive integer $n\not=4^k,4^k\times7,4^k\times13$ $(k=0,1,2,\ldots)$ has the form $x^2+3y^2$ with $x,y\in\mathbb Z$, can one show that $n=x^2+2y^2+3z^2$ for some integers $x,y,z$ with $y\not=0$?

Answer 1

Problems like Question 1 go back to Euler and Gauss. This question is essentially asking which idoneal numbers are sums of two squares. There is a famous open problem going back to Euler and Gauss asking for a complete list of idoneal numbers (equivalently discriminants with one class per genus); the list is known to be finite with one more possible idoneal numbers but this is ineffective. On GRH one would have the Euler/Gauss conjecture.

Similar results hold for these problems. The connection in the context of sums of three positive squares was explicitly fleshed out by Grosswald, Calloway, and Calloway (Proc. AMS 1959, vol 10, 451--455), who formulate a version of Question 1 as a conjecture.