If you have a common cold and on day 7 you have close contact with someone with covid-19 (and at contagious state), are you somehow protected against covid-19 infection/symptoms as your body was already fighting another coronavirus?

A "cold" can have many different sources. including coronavirus, but also adenovirus, metapnuemovirus, RSV, Rhinovirus, etc. When a person is sick, their immune system is ramped up fighting whatever virus they have. There can be some cross-protectiveness, but there may not be. It is wise to avoid contact with ill people when you are ill as you can catch other viruses.