Can international students in USA intern in summer in industry? Can international PhD students go do internships during summer in industries? 

Does their F1 visa status allow this?

From how I see it, this is neither an OPT or a CPT. 

I asked my International Office and they said I could spend a term at another university because then I'm still enrolled Full Time (just at another institution) which I believe is fine with some paperwork. However, they are getting back to me (it's been a while so I am asking here) whether anyone else has done something similar?
Yes you can, and this is exactly what CPT is for. (assuming you get paid for the internship. If you are not being paid - I'm not sure what the rules are, but it might be easier than CPT)

You can do it for up to 12 month (not necessarily consecutive) during your degree. Check again with your international Office for the exact rules (you might still need to be enrolled to 12 units if not summer quarter/semester).