Sorry, my reply was gone nowhere.
A1: no, not wrong. It is display and numerical discrete result.
A2: the data is the original data. You need to extract where the material/geometry begins. for example:
clear;
index=getresult("monitor","index");
ind=index.index_x;
nmax=max(ind);
x=index.x;
y=index.y;
plot(x,ind(:,find(y,0)));
linex=ind(:,find(y,0));
aa=find(linex>1);
?x(aa);
This is all the x data for the material.
index_x,index_y and index_z are for Ex,Ey and Ez from the Yee mesh, since FDTD makes them displaced in purpose. Inside FDTD, the discrete geometry will look differently from real word.
A3: after discritization the geometry will be different. For visualization FDTD has interpolated them into the same location. Please refer the Yee mesh:
A4: the grids you are referring to is for visualization only. It has no any effect for simulation and discritization.