It sounds like Lilith is more of a Persecutor Alter than an Introject.
It can be hard/confusing sometimes to tell the difference, but I found this, so hopefully it's helpful to you.
*edited by admin*Introjects:
They would have been created during a traumatic incident just as any other alter.
Introjects are alters who were split off to represent outside people, most typically an abuser (but not limited to that, by any means), and thus create the appearance of being “introjected” within your system from an outside person. They are splits from your own mind, and they are there to help you remember / contain specific, detailed information related to whoever it is that they are “being” within your system.
Introjects are as convinced as the other parts of the system that they the same as the external people they represent. They think they are separate from the survivors, and separate from the body of the survivor. Many negative introjects will adamantly believe that they could hurt or harm the survivor / host of the system and not be hurt themselves. Introjects typically truly believe they are separate people, but they are, in fact, part of the DID system.
Persecutors:
Persecutor alters can be, and often are the same as the introjects. Some persecutor alters are alters from your system that internalized the rules of the perpetrator, and continue to follow those rules, but don’t necessarily believe themselves to actually BE the same perpetrator person. Introjects actually think they are that perpetrator person.
With your other questions, I wish I could be more help, but unfortunately, we have done no therapy work with our introjects yet, so we aren't sure how to go about it. I agree with Cassandra about the communication, keep the lines open the same way you do with others, but be prepared. The one thing we have noticed with both our introjects and persecutor alters, is they know when we are feeling vulnerable, and they use it against us. So just be careful, and be ready for them to say/do just about anything.
Good luck.