How exactly will capturing Space Dandy win the war for the Gogol Empire?
That is the driving question of the series. It's hard to tell if it will; considering the nature of the show, maybe they're just wrong. Or maybe it has something to do with his ability to ignore death and probability. There's just not enough information to know for sure yet.
It appears to have something to do with Pyonium.
Episode 3: Occasionally Even the Deceiver is Deceived, Baby
If the The Deathgerian was recorded in magazines, how did Dandy get paid for its discovery?
Just because people knew it existed doesn't mean it was properly scanned and cataloged. Considering how scared of it everyone is, it was probably never successfully captured.
The Gogol empire knows about it, but that doesn't mean that anyone else has the same information. As far as we know, Dandy only captures aliens for the human race, and all of the Gogol troops we've seen are non-human.
Also, the picture the Gogol Empire magazine has is of her human disguise. When Dandy presents Mamitas to the ARC, it's in her "true"(albeit shrunken/starved) state.
Episode 8: The Lonely Pooch Planet, Baby
So, if Laika is the same dog from real life, than how exactly did it survive on Planet Machinia for hundreds, if not thousands of years well beyond a dog's natural lifespan.
She could have gone through a wormhole on the way there which sent her forward in time. Alternatively, if Sputnik 2 was going fast enough, relativistic time dilation would have caused much less time to pass for her than for everyone else. Or it could be an unknown interaction between the planet's atmosphere and canine biology.
Keep in mind, it's also entirely possible that it's not the same Laika.
Wasn't it implied that when the alien fleas landed on her body it extended her life-span, but didn't make her immortal?
Possible, but never implied, especially considering they showed no special powers beyond machine manipulation.
They have the power to manipulate space-time, which is why Machina turns into a black hole when they die.
Episode 13: Even Vacuum Cleaners Fall In Love, Baby
What happened to Coffee Maker? She's left completely unscathed even after the Doomsday Device becomes active, and then QT just kind of leaves her at the scrap yard. Did he just decide to let her go?
QT doesn't just let her go, it's a case of I want my beloved to be happy. QT pretty much said so himself moments before leaving Maker: "It was always Register". Not to mention after the end of the the massive fight against the Doomsday Device, QT can be seen floating in the water next to Register.
Episode 21: A World Without Sadness, Baby
Meow and QT, you just almost died, and your first assumption is that Dandy is taking a nap? Why wouldn't you check to see if he's still alive or needs medical attention?
They haven't been shown to be the most intelligent in literally anything, but injuries I'm sure are reserved for an even higher rung of incompetence. Remember when Meow was a Zombie (aka literally dead himself) and they assumed he had a cold?
What sets Dandy apart from the other ghosts, that he can cure Poe's loneliness?
Presumably the fact that he has the potential to be brought back to life (maybe something that relates to the cosmic string), or simply refuses to accept that he's dead.
He is the first new person to come to the planet since it's "death"; everyone else already there has been there since before the end, so Dandy is able to provide new experiences and view that Poe hasn't had in ages.
Episode 24: An Other-Dimensional Tale, Baby
Dandy claims that warping doesn't actually exist. He explains that 'warping' is actually just another you in another universe now thinking that it's you, and apparently that you in another universe is just conveniently where you always want to be, and says that "that's all it is." This leads to some problems.
Taken at face-value this basically means it's just your mind that goes to another universe and there's no actual physical warping. But at beginning of the episode they were just talking about how they had to keep warping to chase down another alien who also kept warping. If warping doesn't actually exist and is actually just mind transference to another universe, why would they have to 'warp' after another warping alien? Wouldn't ANYONE watching someone else 'warp' notice that they're not warping? And where are the people warping IN from somewhere else coming from? So it just being universe mind-transferring with nothing physical involved doesn't make sense.
So if that's out, does that mean their whole bodies and vehicles are actually being sent to another universe? But then how is that not count as warping? Even if you didn't notice you're crossing to a different universe, you're still warping from one universe to another, which still counts as warping. And while the idea that when you warp you're warping to another universe in the place you want to be is feasible, the reverse (the idea that, say, when you warp to a restaurant another you from ANOTHER universe warps to the restaurant you wanted to go to right after you warp out, so nobody thinks you just disappeared from the universe) working out every single time is a bit contrived to say the least.
Episode 25: Dandy's Day in Court, Baby
How did the baseball get exposed to pyonium?
For that matter, they never explained why Dandy was at the victim's house.