Yes, the entity is intact with one member leaving and adding another and can use the same Federal EIN.

No, buying one person's interest in an LLC will not be a like-kind exchange. 

'Like-Kind Property' Any two assets or properties that are considered to be the same type, making an exchange between them tax free. To qualify as like kind, two assets must be of the same type (e.g. two pieces of residential real estate), but do not have to be of the same quality.