Sum of all matrix entries

You can use Total. Here's a matrix of the given size:

m = RandomReal[1, {10000, 10000}];

And here's the output of Total:

Total[m, 2] // AbsoluteTiming

{0.041005, 4.99984*10^7}

Here is another solution that performs similarly to Total:

v = ConstantArray[1, 10000];
m = RandomReal[1, {10000, 10000}];
v.m.v // AbsoluteTiming

{0.040816, 4.99968*10^7}

Total[m, 2] // AbsoluteTiming

{0.041774, 4.99968*10^7}

The reason that these solutions are fast is that they use vectorization. Using Compile is not quite as fast, especially not when compiling loops:

sum = Compile[{{m, _Real, 2}},
   Sum[Compile`GetElement[m, i, j], {i, Length[m]}, {j, Length[m]}],
   CompilationTarget -> "C",
   RuntimeOptions -> "Speed"
   ];

sum[m] // AbsoluteTiming

{0.092583, 5.00016*10^7}

sum = Compile[{{m, _Real, 2}}, Module[{tot},
   tot = 0.;
   Do[
    Do[
     tot += Compile`GetElement[m, i, j],
     {i, j + 1, Length[m]}
     ],
    {j, Length[m] - 1}
    ];
   2 tot + Tr[m]
   ],
  CompilationTarget -> "C",
  RuntimeOptions -> "Speed"
  ]

sum[m] // AbsoluteTiming

{1.1837, 4.99976*10^7}