How can I visualize and calculate the volume of this $3$-dimensional object?

You can try RegionPlot3D:

RegionPlot3D[Sin[u] <= Cos[v] && Sin[v] <= Cos[w] && Sin[w] <= Cos[u],
   {u, 0, 4 Pi}, {v, 0, 4 Pi}, {w, 0, 4 Pi}, PlotPoints -> 90]

Or use ImplicitRegion:

ir = ImplicitRegion[Sin[u] <= Cos[v] && Sin[v] <= Cos[w] && Sin[w] <= Cos[u],
     {{u, 0, 4 Pi}, {v, 0, 4 Pi}, {w, 0, 4 Pi}}];

RegionPlot3D[ir, PlotPoints -> 100]

I question the accuracy of the volume calculated by DiscretizeRegion: Just excise one out of the bunch to get the plot:

First compute the volume via DiscretizeRegion:

ir = ImplicitRegion[
  Sin[u] <= Cos[v] && Sin[v] <= Cos[w] && 
   Sin[w] <= Cos[u], {{u, Pi/2, 3 Pi}, {v, Pi/2, 3 Pi}, {w, Pi/2, 
    3 Pi}}]; Volume@DiscretizeRegion@ir

Results: 32.54

Now do a Monte-Carlo integration series from 5000 to 1000000 points:

myTable = Table[
   myPts = RandomReal[{Pi/2, 3 Pi}, {num, 3}];
   pts = Length@
     Select[myPts, (Sin[#[[1]]] <= Cos[#[[2]]] && 
         Sin[#[[2]]] <= Cos[#[[3]]] && Sin[#[[3]]] <= Cos[#[[1]]]) &];
   pts/num (5 Pi/2)^3 // N, {num, 5000, 1000000, 5000}];

and plot the results and do a Fit of the data:

Fit[myTable, {1, x}, x]

34.9067 -0.000148992 x

which appears to be settling on a value significantly higher than 32.5 computed by DiscretizeRegion.