we know that he wasn't Richardson's VP, therefore, if he posted someone to the Office of Vice President, then that VP would be able to challenge Eden's claim to Acting President, I think I'm not to sure on this one but it fits my profile of the mad bastard so I roll with it.
Quick clarification: Eden would not be acting President as that refers to a vice-president (or other cabinet member) who assumes authority when the President is incapacitated (but still alive). Eden, since Richardson and the rest of the Presidential line are very much dead, has now become the official President.
As to his long term of office, its certainly not necessarily democracy, but in a state of emergency, democracy is shoved aside in favor of national security (we can see that in the real world as well). The 22nd Amendment has presumably been suspended (which Eden, with his broad emergency powers enacted, and no congresss or supreme court to counter him, would be perfectly justified in doing).
Also, any VP he appointed would not be able to challenge his Presidency, I believe earlier you posted a quote that refers to the ability of Congress to appoint a Speaker of the House or President Pro tempore of the Senate who could then challenge a cabinent member President's authority to rule, however, since Congress no longer exists (and this would be the only government body capable of doing this) this rule cannot apply to Eden, nor does it apply to any of his own cabinent that he appoints.