My question involves labor and employment law for the state of:

I was terminated per the paperwork for misconduct with HR saying to me in a follow-up call that I could apply for jobs though it maybe difficult to get re-hired. After working my network and with several calls and written letters to HR and others at this firm, I received a strongly worded letter from this firm asking me to cease and desist all contact including
- my personal contact
- anyone contact them on my behalf
- any contact via any communication mechanism

to obtain re-employment at the firm.

I have spoken with the NLRB as I believe I have a claim.

Questions:
Would contact from the NLRB constitute a contact that meets the criteria of the cease and desist letter?
Would future employers background search show a claim made at the NLRB ?
Would showing a claim at the NLRB hurt my chances for future employment?