From a lawyer:
The answer to question one is that it should be we, because the letter is sent by the law firm to the other law firm, rather than the individual. It is normally we because law firms were traditionally partnerships, so the we technically refers to the partners
On the second one, again the letter is sent on behalf of the firm, but addressed to the person at the law firm, so it would normally be I, but you can also use we, for example if you are asking for documents or information it is the law firm that wants to receive them, rather than the individual.
Student or Learner