Well, there are some cases where a last name can also be a trademarked name I think, like my last name (no, I don't own the trademark...!). But I can and do use my last name in my domain and my business because I don't have anything that would make people think I am the company that owns the trademarked name.
But to answer your question, if the domain is something like personnameDisease.com, then I think you would be pretty safe, generally speaking, as long as someone didn't trademark the disease.
To answer the rest of your question, there are many opinions, but almost everyone agrees that .com is the very best to register. Now, in this case it may depend on what you are going to do with the domain. A drug company may be interested in the .com domain name, but a charity or foundation would prefer the .org version of the domain.
Now that you have registered it can you say what it is?