I am an Advanced OW diver, EANx certified, and getting ready to take my Stress Rescue course in a couple months to become a Master Diver. Like you, I love to build on my experience and education and have already obtained numerous Specialty course certifications.
To answer your Question#1: Depending on the cavern course you take, and diving caves at all for that matter, check first to see what prereqs are required to dive where you plan on going. Cave diving is a completely different world with excess risks involved and knowledge required. Some Florida caves, for example, will not allow you to dive them w/out first having a cave certification and so many logged dives under your belt. You may indeed be more comfortable and enjoy caving better with more dives under your belt, albeit from completing an Advanced OW course, etc. I have not yet logged any cave dives and would not do so until I completed a cave course. As a beginning diver many years ago, there is no way I would have felt comfortable caving w/out having logged a lot more dives, for whatever it is worth. Network with current cavers that have dived where you plan to dive and find out what they recommend, etc.
Question #2: Depending on what organization your Advanced OW course is obtained from, usually being an Advanced OW diver simply means you are more experienced and knowledgeable. There are indeed some dive sites that are recommended for Advanced divers and beyond, and certain dive operators will only take Advanced divers to certain sites, etc. When I dive the Carribean, for example, a couple of the dive operators I have used want to see my C-card, which states Advanced OW, and verify how many logged dives I have so they can take me and other divers with my skill level and comfort level to dive sites we might enjoy more so than, say, beginning divers. Again, some sites and/or night dives are not recommended for beginners but more for Advanced.
Hope this helps!!