1. Are there any states that ONLY allow CACREP accredited training programs? It seems only about half of the states mention CACREP but most accept it as one form of training but allow for many variations.
I haven't heard of states being strict about this, so you'd have to literally look up each state board at the master's level to determine which require CACREP-accredited graduates. CA is one of the toughest states in the country for mental health practitioners regulations-wise, yet still allows graduates from only regionally-accredited programs (i.e non-CACREP) so long as they meet the specific course requirements within the program. If CA is any indication of the standard, CACREP-accreditation isn't required.
2. It appears that psychology is slowly being phased out of masters-level training. How (un)common is it to have programs that are designed for LPC or LPC-like (not Social Work or MFT) training staffed mostly by psychology faculty rather than counseling faculty?
In my doctoral program, we had some classes side-by-side with master's level counselors within the same department, with some courses taught by psychology faculty, the rest taught by adjunct master's level counselors. Not sure how common this is.
3. What exactly is "counseling" rather than "psychology"?
Different foundations entirely, but a lot of overlap. Counseling came from vocational/career counseling and began to incorporate counseling theories into it, psychology came from a science background (i.e. James, Watson, etc.) and incorporated counseling theories into it as well (i.e. Glasser, Rogers, Perls, Ellis, Beck, and so on). I've been in both a counseling master's program and a doctoral psychology program, and there's much overlap in counseling theory and practice, but psychology is heavier on science/research understanding in addition to clinical skills. Counseling doesn't emphasize research/science, generally.
In my psychology training, I noticed I had more exposure and deeper training in more recent counseling theories (ACT, DBT, EFT, Interpersonal Therapy), while my master's program focused on Rogers, Ellis/Beck, Perls, and a lot of the older foundational theories that aren't used much in practice anymore (i.e. Person-Centered Therapy, REBT/CT, Gestalt). Not sure of that was a fluke within my master's program, but it was concerning. I think many master's programs I've heard about at least near me now (on the West Coast) offer a more updated education than my midwest program, but you just never know unless you ask current students or faculty what theories they teach. That is very important to find out prior to going into a master's program.