I couldn't really think of current Eclipso doing anything impressive but the tier thread has her listed as sub-skyfather level so I wasn't sure if I just missed something.
Eclipso would normally be above Skyfather,me thinks.
Eclipso goes up against the Spectre regularly with the fights only ending when Spectre traps Eclipso.
Vaguely. The first half is a Dr. Fate series that isn't connected to Countdown at all.
The second half is this Eclipso story that seems loosely tied to Countdown, except for the reveal that Eclipso is in fact a creation of Darkseid (Or at least the Black Diamond is).
Because the Dr. Fate series was commissioned and written as its own ongoing series, and then got relegated to a Countdown title because of various internal problems (The original artist was overbooked and couldn't do the work, Gerber is terribly ill and working slowly, the marketability of a Dr. Fate book was called into question).
I don't see how Dr. Fate is going to tie into Countdown at all, at least not the direct events of the book.
He actually appears on the last page of the first issue, and his cronies the Furies pluck Eclipso out of the sun's rays during that first issue story as well.
I also think that the Fate story is EXTREMELY strong, so even if you don't usually like Fate I think you should give it a try.
And if you're a plastic man fan, the Eclipso story should be of serious interest to you.
Both of the Countdown to Mystery issues have been thoroughly interesting and entertaining. I would say they are easily among the best new comics I have picked up this year.