I doubt that Crytek can claim anything regards Lumberyard, as it sounds like they signed over all rights to it to Amazon. However, that's really unclear.
It started out a point-version above that which CIG branched from. It's quite a lot further along as Amazon have been making continual updates to it. CIG have already stated that they based it on the same branch of Lumberyard as they were using in CryEngine, and pull in some of the newer features as required.
See above: totally unclear as to how the inclusion of Crytek code in Lumberyard would affect an injunction.
It depends. Let's assume that Crytek kept the source for CryEngine in an SCM, we'll use git as an example. If so, it contains all the history of all the files including the version that CIG branched from. Now how easy the port is depends on whether Amazon were also given the history of the source code. If so, then it's trivial. Assuming they start at commit n, and are now at commit n+10000 they have two choices: 1) they can generate a single patch that encompasses all those commits; 2) they can generate patches for each commit. They switch to commit n of the Lumberyard engine (which will be the same as commit n of CryEngine) and apply their patch(es). Voila, they've successfully migrated to Lumberyard. Their SCM might even make the patching step trivial.
If there's a requirement that to be officially under the "Lumberyard" banner they need to be on the initial Lumberyard point release, at this point they would merge the changes into their repository and manually resolve any conflicts that arise.
If they *don't* have the history, it's a bit more of a headache. First, CIG would revert back to commit n of CryEngine. They would then copy across the initial Lumberyard commit, and finally merge in all the changes they made in their CryEngine branch. If they're lucky, none of what they did will touch any of the same files that were updated in the Lumberyard point release. If they're unlucky, some will and they might have to do a bit of manual conflict resolution. It's probably not a huge job.