3D Image co-registration between multiple subjects
I'm not sure exactly what you mean by the term "dataset", but I will assume you are asking about co-registering multiple images from different patients (i.e. multiple 3D images per subject).
To answer your questions:

If there are no obvious choices about which image is best, then a random choice is fine. If you have e.g. a CT and an MRI for each subject, then co-registration using the CT images is likely going to give you better results because of intrinsic image characteristics (e.g. less distortion, image value linked to physical quantity).
I suppose that depends on what you want to do, but if it is important to have all imaging data in the same co-registered reference space then yes.

Another option is to try and generate an average image, and then use that as a reference to register other images to. Without more information about what you are trying to achieve it's hard to give any more specific advice.
When preforming image co-registration of multiple subjects, how should we select the reference image?

Can a randomly selected image form one dataset could be the reference image for an image from the other dataset?
If we do that, should all the images belonging to the reference image dataset be co-registered with the reference image as well?

I couldn't find any material in this area. Could someone please advice?