I'm hoping I follow you correctly, but it may help to do a screener of your Mass Properties window.
1. Is the pink coordinate the COM? As far as I recall, the coordinate with red X, green Y and blue Z (appears after choosing Tools>Measure) is the origin of the part or assembly.
Yes, it is the location of the COM, and the RBG CSYS is the part origin.
2. On each pink coordinate axis, there is a small line. I noticed that when I rotate the part, these small lines are not parallel to the corresponding axis. Does that mean anything?
Those lines correspond to the principal axes of inertia. In a perfectly symmectical part they will line up, but most often not.
3. When opening a file, I am asked if I wish to run Import Diagnostics. If I say no, a window called FeatureWorks shows up and asks "Do you want to proceed with feature recognition? Do my answers to these questions affect the mass, COM and Moment of Inertia? As somebody else created the file, I have no idea how the proper real part is like. So, it is difficult for me to decide how to answer the questions of Import Diagnostics.
You are opening a model not created in SW, so it will try to assign features such as extrudes, fillets, etc. to the imported solid, other than just a lump solid. This is to try and allow some degree of editability.
4. In Mass Properties, what does "Output coordinate system: -- default ---" means?
This is what CSYS you want to evaluate the mass properties around. For instance you can create a reference coordinate system at the front and center of your part, then evaluate the x,y,z distances to the COM, as well as the moments of inertia about that reference point.
5. On the top left (feature tree), I see "Material <not specified>". Does that mean the mass, COM, and Inertia reported by Tools>Mass Properties are incorrect? How come the person who created the CAD model did not specify the material? Without such information, how can I obtain the correct data from the CAD file?
I'm not 100% on this, but SW references local materials files for reference, and assigns a default value if none is found that it recognizes. You would need to assign one either based on actual part info or contact the vendor for specifics.
6. Although the values of Principal axes of inertia and principal moments of inertia (taken at the COM) are not zeros, the values of Moments of Inertia (taken at the COM and aligned with the output coordinate system) and that (taken at the output coordinate system) are all zeros. Is there somehting wrong? What does this mean?
This is where a screenshot might help. Again it all has to do with what material used, and what your output CSYS is, but my thought are they should not be zero and it is related to your not having a material assigned.
Hope this helps some!
Justin did a nice job of answering most of your questions. When SolidWorks asks you "Do you wish to run Import Diagnostics?" it is a good idea to say yes. This will check your imported part and attempt to repair it if it finds problems. Even if it can't fix them all, it is better to know that your part has errors.
Like Justin, I'm not sure, but I think that SolidWorks applies material properties from your default template. Unless you have set it up differently, I believe that it assumes a specific gravity of 1 for the material.