To start, you are are describing Imajn, which will be in open beta soon (http://imajn.com
1. In MultiCore does module = core? Is it bad design to have a module that is not a core? I'm kind of unclear as to what a "core" is. Is it an application? Is it a piece of an application, and if so, is it expected to have a lot of communication with other cores or be fairly independent?
A core is the main application that loads the modules as well as the modules themselves. This is why we don't just say module; the main app is a core as well. As for what each module does and what its communication patterns and protocols are is up to you. (Imajn provides a container and defines several types of cores with different purposes).
2. Is my implementation abusing MultiCore because there is no main application for each core in Tier #2?
It's like Legos - there is no wrong way to put them together.
3. If I went with Solution #2, how would I tell the core what module to load up? (is url params the only way)
All up to you. You might use a Mediator to load and cache these Modules. They are typically components of the view. (Imajn provides this container and loading mechanism, reducing your work to defining a loaded xml configuration.
4. What is in a module? If I have multiple modules that use the same proxie does each of their SWF files contain that proxy? Can/How do you specify what classes are associated with a particular module?
Best not to pass around proxies, but if you must, be sure they are defined in a common package for reuse and don't make one module depend on another in order to reach that class.