1) How does Flex work with ActionScript? Is there an ActionScript editor inside the Flex Builder app?
When working with Flex, you use MXML and ActionScript, but after compilation it all gets compiled into ActionScript.
Flex Builder has an ActionScript editor, but it really just a text editor. Open an .as file and edit it.
2) I want to import projects I developed from Flash into Flex Builder but they have timeline and ActionScript components - is this even possible?
You cannot just import stuff you've done in Flash into Flex, and certainly not as a Flex project. But you can work in Flash and use what you have created in Flex. The Flex SWFLoader component allows you to load a SWF created in Flash and click its buttons, etc. access its data etc. Watch out though as there can be issues with versioning and security.
And then there is always Flex Component Kit for Flash.
3) Will objects I have layed out in Flash also be editable in Flex Builder?
No, you can access them and their data, but you can't edit them. You might be able to add components to containers in a Flash generated SWF, but it is not seamless by any means.
4) Finally, should I just re-do the entire app in Flex?
Hard to say. It is probably just a good idea to rebuild it in Flex, because if you are using snippets of ActionScript throughout the timeline, well that's why Flex came to be, Flash creates timeline based application nightmares, Flex is event based, which is what you want when creating real applications.
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Thank you kind sir for the useful information and relevant links!