initiate proceedings
Question 1:
Dissolution occurs when any partner (or partners) initiates proceedings to terminate the partnership or when an event in the partnership agreement occurs.
What does "initiate proceedings" mean?

Question 2:
Laura Ingraham, a paid Fox News contributor, is in talks to......
What does "contributor" mean in this particular business?

Question

initiate proceedings
Question 1:
      Dissolution occurs when any partner (or partners) initiates proceedings to terminate the partnership or when an event in the partnership agreement occurs.
      What does "initiate proceedings" mean?

Question 2:
      Laura Ingraham, a paid Fox News contributor, is in talks to......
      What does "contributor" mean in this particular business?

Answer

Question 1:
In this context, "initiate proceedings," means that any partner/partners can start the process of terminating the partnership. In other words, to "initiate proceedings" roughly means to start a process.

Question 2:
In this context, "contributor" means a contributor of information, as in Laura Ingraham gives information to Fox News (and is paid for it).

I hope this helps :)