Is a partnership agreement good in the whole country or does each agreement need to be signed in each state?

A partnership valid in one state would be entitled to conduct business in other states. However, there are many potential issues that you should discuss with your CPA. For example, the partnership may need to file tax returns in other states (in which it conducts business); it may need to file with the Secretary of State in the other states (in which it conducts business). Again, those are only examples.