I've 3D printed my brain - I'm a biomedical engineer, and for a while was working for a research institute in France. While there, I had many friends and acquaintances who were doctors or surgeons performing their own research work. I had made noises in the past that I really wanted to print my brain sometime, so one of them referred me to an observational study in the area using fMRI to observe the brain performing visuo-spatial tasks. I went through the study, got paid €50 (score!) and asked for the morphological data in DICOM format, which they happily handed over. The DICOM is essentially a whole bunch of separate images, one for each slice the MRI takes, along with the metadata that helps piece it together. The reconstruction of a 3D model from slice data is usually called segmentation. I'm not overly familiar with the process, but I'm sure someone here can chime in with various segmentation methods. In my case I asked a technical medicine intern to do it for me. Interns can be useful, after all. AFAIK the output is usually a mesh, like an .obj or .stl file, which is thankfully pretty compatible with 3D printing.
To answer your specific points - I would go for MRI over CT - CTs tend to have high radiation doses, especially if you plan to do a full-body scan. I can't recommend it. MRIs are much less harmful. There are private clinics where you can pay for a full body "preventive" MRI - this will set you back a nice sum, >€1000 in the Netherlands, IIRC. Segmenting the resulting DICOM file is difficult. It will require a fair bit of work. Look into ITK / VTK / 3Dslicer. You will absolutely have to clean up and probably remesh any resulting stl/obj, I recommend using meshlab for that. From there, 3D printing can be done with any personal printer that has a large enough volume to not take forever. Using an external service, e.g. shapeways will be very expensive. If you don't have your own printer, you should probably look into an inexpensive FDM printer, I personally use an original prusa i3 Mk2.5.
All in all, it will be a very expensive journey if you're not already in the field and have connections. Maybe try to connect with local universities to find observational studies.
edit because I realised I sound very negative. I actually think you have a really cool idea. Please do go ahead if you have the time and money, and for the love of all that is holy, blog it for posterity!
To answer your specific points - I would go for MRI over CT - CTs tend to have high radiation doses, especially if you plan to do a full-body scan. I can't recommend it. MRIs are much less harmful. There are private clinics where you can pay for a full body "preventive" MRI - this will set you back a nice sum, >€1000 in the Netherlands, IIRC. Segmenting the resulting DICOM file is difficult. It will require a fair bit of work. Look into ITK / VTK / 3Dslicer. You will absolutely have to clean up and probably remesh any resulting stl/obj, I recommend using meshlab for that. From there, 3D printing can be done with any personal printer that has a large enough volume to not take forever. Using an external service, e.g. shapeways will be very expensive. If you don't have your own printer, you should probably look into an inexpensive FDM printer, I personally use an original prusa i3 Mk2.5.
All in all, it will be a very expensive journey if you're not already in the field and have connections. Maybe try to connect with local universities to find observational studies.
edit because I realised I sound very negative. I actually think you have a really cool idea. Please do go ahead if you have the time and money, and for the love of all that is holy, blog it for posterity!