I'm a mutt so I looked into race/ethnicity reporting and read this top-law-schools article: http://www.top-law-schools.com/urm-applicant-faq.html
It claims you should only claim something if you're 1/4 that (Native Americans excluded), which is what I followed. I don't know if these are set rules or just guidelines?
Personally I think you should only claim what you truly consider yourself--application advantages aside. If you are/were close to your PR grandfather and believe you retain significant cultural/racial/etc identities, go for it. When I was considering this, I asked myself if I would claim that race/ethnicity to a group of people who were 100%. For example, would you tell your friends from PR that you significantly identify as PR? Would your family identify as PR? It's obviously up to you and regardless of what you decide, nobody's going to question your reporting changes. Schools don't have the resources to compare and ethnicity is already sensitive as is. In your case you can even say you didn't know what to put because you are two different Hispanic types.
For the immigration question--does your father now have citizenship? Do you? I doubt a law school will take action on an illegal immigrant story. There are many illegal immigrants in law school. They especially won't if you are now citizens. If you feel uncomfortable, you could write a statement that includes insinuations without directly admitting to the illegality. Also, I don't know your personal diversity story, but if it comes from the immigration and is not directly related to the illegality, just don't mention it was illegal. Schools will not research your family's status. In diversity statements it might be good to focus on your own experiences instead of your parents anyway.
Finally...I wouldn't worry about what you put on the exam or LSAC. I think those really get you fee waivers at best. What you report on your school apps is the real deal. Be careful because different schools have different categories so what autofills isn't always correct. I noticed it kept screwing up my stuff from app to app.
Good luck!