Thank you for your question today, I look forward to assisting you. I have nearly 20 years of legal experience in various disciplines, including JAG.
1. Yes, your previous command can note the incident even though you are not in their chain of command. You were in their chain of command when the incident occurred, or at least, when the incident that they are alleging occurred. They will, however, need the cooperation of someone in your current chain of command in order to do anything more than simply documenting the incident.
2. Yes, you'll get a chance to speak to the allegations in some forum. For the incident to have any effect on your career, it has to go into some sort of permanent file, be that a performance report, Letter of Reprimand, or some greater level of legal action. You aren't entitled to speak to the issue until they actually choose a course of action and then, that is when your ability to rebut their statement takes effect.
3. No, just because you are processed that does not mean you'll be discharged. With your career, you would legally be entitled to a board hearing before you could be considered for discharge. You would have representation by counsel and the opportunity to rebut any evidence in a trial-like process before you could be discharged.
4. A board of inquiry or court of inquiry, under Article 138 of the UCMJ, is just a formal process for you to question the actions of command. However, the disciplinary process and investigatory process of a command are not subject to that sort of inquiry. This is because any disciplinary action to be taken comes with it's own due process. A board of inquiry simply provides the means for a person to question discretionary practices of a command where the decision don't have their own due process built in.
For instance, you could do a court of inquiry concerning how people were selected for certain schools that would help their career. You can't do it for how a Captain's Mast was processed though, because that process has its own due process built in.