Why does the Slider value change from an Integer to a Real, causing the Manipulate expression to break?

It seems like a bug. Here's fix, via explicitly constructing the slider (Manipulator):

n =.;
DynamicModule[{c, t, main, f},
 Manipulate[
  ControlActive[{x, y}, main[x, y]],
  {{x, c/2, "n1"}, 1, y - c, 1},
  {{y, n - c/2, "n2"}, x + c, n, 1, Manipulator[#1, {x + c, n, 1}] &},
  SynchronousUpdating -> False,
  Initialization :> (
    n = 300;
    c = 100;
    main[x_, y_] := main[x, y] = f[x, y];)]]

This isolates, or at least localizes the problem to having a Dynamic lower limit to the slider; the argument x + c automatically has Dynamic[] applied to it by Manipulate to make the slider limits dynamically dependent on x and c:

n =.;
DynamicModule[{c, t, main, f},
 Manipulate[
  ControlActive[{x, y}, main[x, y]],
  {{x, c/2, "n1"}, 1, y - c, 1},
  {{y, n - c/2, "n2"}, x + c, n, 1, Manipulator[#1, {Dynamic[x + c], n, 1}] &},
  SynchronousUpdating -> False,
  Initialization :> (
    n = 300;
    c = 100;
    main[x_, y_] := main[x, y] = f[x, y];)]]

Further isolation:

n = 300;

DynamicModule[{y = 200, c = 100, x = 50},   (* fails with Dynamic 1st argument *)
 {Manipulator[Dynamic[y], {Dynamic[x + c], n, 1}], Dynamic@y}
 ]

DynamicModule[{y = 200, c = 100, x = 50},   (* works with no Dynamic argument *)
 {Manipulator[Dynamic[y], {x + c, n, 1}], Dynamic@y}
 ]

DynamicModule[{y = 200, c = 100, x = 50},   (* works with Dynamic 2nd argument *)
 {Manipulator[Dynamic[y], {x + c, Dynamic@n, 1}], Dynamic@y}
 ]