Wart,
To answer your question simply: Yes, Cave 1 has similar standards to Intro. Both are main line navigation only with approximately 1/6th gas rule. Cave 1 allows you to navigate 1 permanent intersection such as a "T" or "Y" in the gold line. Essentially, they are your cave diving "learner's permit." Both have tons of great dives and allow you to gain valuable experience while keeping you within a relatively safe box, so to speak, with loads of gas and no complex navigational decisions.
Fundamentals and Cavern also have similarities in that they both focus on buoyancy, trim, non-silting precision kicks, and teamwork. As you mentioned, you'll need to take fundamentals in doubles to achieve a "tech pass" to move on to either Cave 1 or Tech 1. This just assures that you go into the next level classes with a solid foundation and are able to perform valve drills, s-drills, hose management, etc while handling mild task loading and maintaining good buoyancy and control. This way, you aren't spending your time working on those skills when you could be cave diving! A difference between these 2 courses is that fundies does not have any overhead training, while cavern obviously introduces you to the "daylight zone."
Full cave is more or less equivalent to GUE Cave 2. It allows you to start doing jumps, circuits, traverses, gaps, siphons, 1/3 gas rule. I'm not sure about the specific IANTD standards of what Full cave includes, but GUE Cave 2 allows you all of the things just mentioned plus deco and stages. A couple of agencies (NACD, NSS-CDS) also have an "Apprentice" level cave diver which I believe allows you to do 1 jump plus 1/3 gas rule. Someone chime in and correct me if that is wrong. Basically it's an intermediate level to afford more experience and planning before going onto Full cave. It would be equivalent to GUE "Cave 1.5" if it existed.
I also don't think that any of the agencies allow scootering during Full Cave and require extra training for it.
Hope that helps and answers some of the questions you had!
Kyle