Theorems V in [this paper][1] of [L.E. Dickson][2] states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \  \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \in \mathbb{N}\}.$$
Let $\mathbb{A}$ denote $\mathbb{N}$ or $\mathbb{Z}$. Consider the following set:   $$E(\mathbb{A}) = \left\{\frac{1}{2}\|u+v \|^2 \text{ with } u,v \in \mathbb{A}^3 \text{ and } \|u \| = \|v \| \right\}.$$

Let $u= (a,b,c)$, $v = (b,-a,c) \in \mathbb{Z}^3$. Then $\|u \| = \|v \|$ and $\frac{1}{2}\|u+v \|^2 = a^2+b^2+2c^2.$  
It follows that $F= E \subseteq E(\mathbb{Z})$.  Now, by [Legendre's three-square theorem][5], $E(\mathbb{Z}) \subset F$ also.   
Then, we have an extension of Dickson's theorem as $E(\mathbb{Z}) = F$. Now, what about $E(\mathbb{N})$? 

Take $u=v \in \mathbb{N}^3$, then $\frac{1}{2}\|u+v \|^2 = 2 \|u \|^2$, so by Legendre's three-square theorem, $E(\mathbb{N})$ contains the even part $F$. The computation below shows that $E(\mathbb{N})$ contains every odd number less than $95362$, except those in $I=\{ 5, 23, 29, 65, 167 \}$, suggesting that $E(\mathbb{N}) = F \setminus I$. 

> **Question**: Is it true that, for $u,v \in \mathbb{N}^3$ with $\|u \| = \|v \|$, the form $\frac{1}{2} \|u+v \|^2$ covers every odd
> number, except those in $\{ 5, 23, 29, 65, 167 \}$?

*Application*: [this answer][4] proves that the form $\| A\|^2$ covers every natural number for $A \in M_3(\mathbb{Z})$.  
 A positive answer to the above question would prove this result for $A \in M_3(\mathbb{N})$.  

___
For the convenience of the reader, the answer of Philipp Lampe (of what was **Question 1** in a previous version) was incorporated in the post.
___
*Computation*   

    sage: L=cover(135)
    sage: set([2*i+1 for i in range(47681)])-set(L)
    {5, 23, 29, 65, 167}

____
*Code*

    # %attach SAGE/3by3.spyx
    
    from sage.all import *
  
    cpdef cover(int r):
    	cdef int a1,a2,a3,b1,b2,b3,x,n
    	cdef list L
    	L=[]
    	for a1 in range(r):
    		for a2 in range(a1+1):
    			for a3 in range(a2+1):
    				x=a1**2+a2**2+a3**2
    				for b1 in range(isqrt(x)+1):
    					for b2 in range(isqrt(x-b1**2)+1):
    						for b3 in range(isqrt(x-b1**2-b2**2)+1):
    							if a1**2+a2**2+a3**2==b1**2+b2**2+b3**2:
    								n=((a1+b1)**2+(a2+b2)**2+(a3+b3)**2)/2
    								if is_odd(n) and not n in L:
    									L.append(n)
    	return L


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=1561323
  [2]: https://en.wikipedia.org/wiki/Leonard_Eugene_Dickson
  [3]: https://en.wikipedia.org/wiki/Dot_product
  [4]: https://math.stackexchange.com/a/2967989/84284
  [5]: https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem