How can Planetary be part of the same universe as The Authority? If the Four are supposed to be hiding and/or destroying all that fantastic things that could stand a chance against their evil expies of everything ever drawn by Jack Kirby, then shouldn't they do something about those totalitarian superheroes who overthrew the US government and killed God?
The Authority are either too powerful to take down or are allowed to exist because they do wonderful, helpful things like turn the US into a police state (give the conspiracy theorists a bigger target) and kill God (definitely a superhuman force that the Four couldn't control or conquer, so had to be eliminated). Imagine this - you're a group of four posthumans who are very, very good at getting rid of other superhumans. And you want to take over the US. But the US is run by a lot of bureaucrats and elected officials and such, and killing them all would draw a lot of attention. So you let a group of less than ten people take over the whole place. Toppling or subverting this regime is a lot easier. Or, as we learned in the same issue of Planetary that introduced the Four, the real rulers of America are thirty-three levels above the president, so the Four don't really give a shit because they're still the Man Behind the Man (repeated 32 times). Or they could have been dead by then. I'm not quite sure how Planetary's timeline syncs up with the post-Jenny Sparks Authority's timeframe.
The Planetary/Authority crossover has Jenny still alive, and Snow is still not at full mental function (ie the blocks Dowling put on him are still present), so that's before issue #11 of Planetary and issue #8 (IIRC) of Authority.
It's repeatedly implied that the Four are just that damned arrogant. They probably consider themselves above all that superhero crap.
Jenny Sparks spends roughly the same amount of time that Snow spent in the desert in a drunken stupor. It's not too great a leap to assume that Randell Dowling got to her as well, thus neutralizing the last two Century babies and was bankrolling the post-"screw you America" Jacob Kreigskind in case either Snow or Sparks got back into the game to keep them busy
I always took it to be a simple matter of timing. Yeah, Planetary took 10 years to make, but the events of the book from start to finish may only take a few months. The Authority doesn't really get going until after Planetary has rescued Axel Brass and Elijah has his memories back, so I saw it as The Four wanting to get rid of Planetary before worrying about anyone else. And besides, there's no indication that Jenny/The Authority knew anything about The Four.
The official take on the pre-DC-merger "Wildstorm Universe" is similar to that of the Image superhero universe, in that all titles are assumed to be in their own universes until such a time as it's indicated otherwise. Thus, while Planetary fits rather neatly into the quiet Cold War environment of late Stormwatch, it doesn't play ball very well at all with post-Ellis Authority even before the world got blown up.
Jakita expressing surprise that aliens have visited Earth, when at least some of those visits are public knowledge in the Wildstorm verse. Warren Ellis doesn't even have the excuse of ignorance, since he's written rather extensively in that universe.
My personal WMG for this and the above is simply that Planetary in fact takes place in a universe which is very similar to-in-many-ways-but-not-quite the actual Wildstorm universe. It's just easier that way.
She also seemed more surprised at the amount of aliens who've visited Earth than at their existence.
Radioactive Half-life Doesn't Work That Way: Why would a radioactive woman "with a half-life of 50 years" stay fully functional for exactly fifty years and then instantly blink out of existence?
More specifically, Planetary takes place in a universe that is pretty much made of various tropes of real world fiction, progressing historically along with our fiction. In the 1920s and 30s, Snow lived pulp adventures just like were written in the 1920s and 30s. The woman was a 1950s Cold War mad science story, so she would by necessity obey the presentation you'd get out of schlocky, scientifically-illiterate period SF written by a hack. Her "story" literally obeys different rules than the turn-of-the-millennium main storyline.
I seem to recall Warren Ellis openly admitted that he knew half-life didn't mean that, but the metaphor of a 'half-life' given from the science was too good not to use.
In issue 17 "The Lost City of Opak-Re", Elijah is attacked by the nominative city's guard- a cyborg-sea-monster-dinosaur-thing. He opens fire on it with his revolver and does some damage. The monster then knocks Elijah off his feet. Elijah loses his gun in the fall, and then acts as if he is completely defenceless. He is only saved by the sudden appearance of a Tarzan-Captain Ersatz. Did Elijah just forget that he can freeze things solid by thinking at them? Even if he did have a gun, wouldn't it make more sense to start the fight using the superpowers that had previously let him Curb Stomp Dracula rather than rely on a mere handgun?
He's visibly exhausted from his jungle trek at the start of that scene, and given that he's already lost the rest of his expedition and bearers he might also have been out of food. So he might not have had the strength to use his powers at full blast, and just giving the giant snake an ice cream headache wouldn't have had any point.