05AB1E, 45 38 bytes

gn¹à@¹˜āsKœ0ªεΘr.;¹gôD©ø®Å®Å/)O˜Ë}à*

Also uses $0$ as placeholder. The more $0$s in the input, the slower the program is. Size of the matrix doesn't matter that much (a 10x10 matrix with three $0$s runs quite a bit faster than a 3x3 matrix with seven $0$s).

Could have been 4 bytes less, but there is currently a bug in the builtin .; with 2D lists. : and .: work as expected, but .; doesn't do anything on 2D lists right now.. hence the work-around of ˜ and ¹gô to flatten the matrix; use .; on the list; and transform it back into a matrix again.

Try it online or verify some more test cases. (NOTE: Last test case of the challenge description is not included, because it has way too many 0s..)

Explanation:

g          # Get the length of the (implicit) input-matrix (amount of rows)
           #  i.e. [[8,0,6],[0,5,0],[0,0,2]] → 3
 n         # Square it
           #  → 9
  ¹        # Push the input-matrix again
   à       # Pop and push its flattened maximum
           #  → 8
    @      # Check if the squared matrix-dimension is >= this maximum
           #  → 9 => 8 → 1 (truthy)
¹          # Push the input-matrix again
 ˜         # Flatten it
           #  → [8,0,6,0,5,0,0,0,2]
  ā        # Push a list in the range [1,length] (without popping)
           #  → [1,2,3,4,5,6,7,8,9]
   s       # Swap so the flattened input is at the top of the stack again
    K      # Remove all these numbers from the ranged list
           #  → [1,3,4,7,9]
œ          # Get all possible permutations of the remaining numbers
           # (this part is the main bottleneck of the program;
           #  the more 0s and too high numbers, the more permutations)
           #   i.e. [1,3,4,7,9] → [[1,3,4,7,9],[1,3,4,9,7],...,[9,7,4,1,3],[9,7,4,3,1]]
 0ª        # Add an item 0 to the list (workaround for inputs without any 0s)
           #  i.e. [[1,3,4,7,9],[1,3,4,9,7],...,[9,7,4,1,3],[9,7,4,3,1]] 
           #   → [[1,3,4,7,9],[1,3,4,9,7],...,[9,7,4,1,3],[9,7,4,3,1],"0"] 
ε          # Map each permutation to:
 Î         #  Push 0 and the input-matrix
  ˜        #  Flatten the matrix again
   r       #  Reverse the items on the stack, so the order is [flat_input, 0, curr_perm]
    .;     #  Replace all 0s with the numbers in the permutation one by one
           #   i.e. [8,0,6,0,5,0,0,0,2] and [1,3,4,7,9]
           #    → [8,1,6,3,5,4,7,9,2]
      ¹g   #  Push the input-dimension again
        ô  #  And split the flattened list into parts of that size,
           #  basically transforming it back into a matrix
           #   i.e. [8,1,6,3,5,4,7,9,2] and 3 → [[8,1,6],[3,5,4],[7,9,2]]
 D         #  Duplicate the current matrix with all 0s filled in
  ©        #  Store it in variable `®` (without popping)
   ø       #  Zip/transpose; swapping rows/columns of the top matrix
           #   → [[8,3,7],[1,5,9],[6,4,2]]
 ®Å       #  Get the top-left to bottom-right main diagonal of `®`
           #   i.e. [[8,1,6],[3,5,4],[7,9,2]] → [8,5,2]
 ®Å/       #  Get the top-right to bottom-left main diagonal of `®`
           #   i.e. [[8,1,6],[3,5,4],[7,9,2]] → [6,5,7]
    )      #  Wrap everything on the stack into a list
           #   → [[[8,1,6],[3,5,4],[7,9,2]],
           #      [[8,3,7],[1,5,9],[6,4,2]],
           #      [8,5,2],
           #      [6,5,7]]
     O     #  Sum each inner list
           #   → [[15,12,18],[18,15,12],15,18]
      ˜    #  Flatten it
           #   → [15,12,18,18,15,12,15,18]
       Ë   #  Check if all values are the same
           #   → 0 (falsey)
}à         # After the map: Check if any are truthy by taking the maximum
           #  → 1 (truthy)
*          # And multiply it to the check we did at the start to verify both are truthy
           #  → 1 (truthy)
           # (after which the result is output implicitly)

The part D©ø®Å®Å/)O˜Ë is also used in my 05AB1E answer for the Verify Magic Square challenge, so see that answer for a more in-depth explanation about that part of the code.

JavaScript (ES6), 270 268 bytes

Takes the matrix as a 2D array. Returns 0 or 1.

a=>(g=(x,y=0,w=a.length,p,R=a[y])=>[0,1,2,3].some(d=>a.some((r,y)=>(p=s)^(s=r.reduce((p,v,x)=>(o|=1<<(v=[v,(b=a[x])[y],b[x++],b[w-x]][d]),p+v),0))&&p),s=o=0)||o/2+1!=1<<w*w?R&&[...Array(w*w)].map((_,n)=>(p=R[x])==++n|!p&&(R[x]=n,g(z=(x+1)%w,y+!z),R[x]=p)):r=1)(r=0)&&r

Test cases

This is definitely too slow for the last test case. :-(

let f =

a=>(g=(x,y=0,w=a.length,p,R=a[y])=>[0,1,2,3].some(d=>a.some((r,y)=>(p=s)^(s=r.reduce((p,v,x)=>(o|=1<<(v=[v,(b=a[x])[y],b[x++],b[w-x]][d]),p+v),0))&&p),s=o=0)||o/2+1!=1<<w*w?R&&[...Array(w*w)].map((_,n)=>(p=R[x])==++n|!p&&(R[x]=n,g(z=(x+1)%w,y+!z),R[x]=p)):r=1)(r=0)&&r

console.log(f([
  [ 2,  2 ],
  [ 2,  0 ]
]));

console.log(f([
  [ 8,  0,  6 ],
  [ 0,  5,  0 ],
  [ 0,  9,  2 ]
]));

console.log(f([
  [ 16,   2,   3,  13 ],
  [  5,  11,  10,   8 ],
  [  9,   7,   6,  12 ],
  [  4,  14,  15,   1 ]
]));

console.log(f([
  [ 10,  0,  1 ],
  [  0,  5,  9 ],
  [  3,  7,  5 ]
]));