When doing coregistration the transformation is actually stored in the header of the source image.
If you want to reslice (interpolate) your source image to also have the same image dimensions and orientation as your reference image, you should 'coregister&reslice'. Then a new image is generated (r+source image)
Indeed, source and reference images need to be the exact same dimensions with exact same voxel sizes. Depending upon the software used to display the 3D rendering, the origin (0,0,0) often needs to be the same as well. I have always found InsightSNAP to produce the best renderings. My book (jeromemallershandbookofstructuralbrainmrianalysis.yolasite.com/ ) shows how to do this in Section 11.