Sure, here are the answers filled in with the correct relative pronouns, indicating whether they are subjects (S) or objects (O), and whether they can be omitted or not:

1. That's the blouse **which/that** I bought last week. (O - can be omitted)
2. My best friend is a person **who** loves reading. (S - cannot be omitted)
3. The dog **that/which** bit me lives next door. (S - cannot be omitted)
4. Is this the dress **that/which** you borrowed from your sister? (O - can be omitted)
5. Do you know the name of the person **who** discovered America? (S - cannot be omitted)
6. The documentary **that/which** I saw last night was about nature. (O - can be omitted)