You can't really define anesthetists (whether AA or CRNA) by the types of cases that they do. That's generally up to the individual practice and medical staff policies, usually through the credentialing process. That same process applies to physicians. Most docs essentially have an unrestricted license to practice in a given state. That's why dermatologists can claim to be "cosmetic surgeons" and why family practice guys can hang out a "pain management" shingle. However, in a hospital, you have to be credentialed for each procedure you do - if you're not credentialed, you can't do it.
So - to answer your questions - AA's of course are always working with medical direction, regardless of whether they're doing liver transplants or cataracts. CRNA's may be working under medical direction, medical supervision, or independently. The opt-out concept refers to Medicare billing practices only. CRNA's can work without anesthesiologists in every state - whether they can work with or without physician supervision is a matter decided by state law or regulation. The bigger/more complex the case, the more chance that there is an anesthesiologist involved at some level. Again, that's up to the individual hospital or practice. In some places, anesthesiologists do all the cases. In some places, anesthesiologists do all the big cases (hearts, neuro, etc.) But also in many places, both AA's and CRNA's are doing hearts, transplants, chest, neuro, complex peds, etc. But in the vast majority of those places (and of course all of them if involving an AA), an anesthesiologist is still running the show. Some of my online CRNA friends tell me they are doing these cases independently. I take that with a big grain of salt, and if they are, those situations are few and far between. Ask ANY surgeon doing hearts and neuro and major peds and transplants (and a lot of other types of procedures) if they would like to be doing their cases with NO anesthesiologist involved or available at any point. That answer would be universally NO. The CRNA's certainly don't agree with that statement, but surgeons strongly prefer that "peer to peer" relationship that comes from working with anesthesiologists - it's simply not going to be there when they're dealing exclusively with a nurse anesthetist.