This occurs in both Godzilla vs King Ghidorah and Final Wars. The heroes release Godzilla to fight the villain's monster or monsters, and then the villains stupidly order their monster to go kill Godzilla. Why? It seems that Godzilla in these instances is not friendly to humanity and in both cases the villains just want to cause the destruction of major cities. They should have done the opposite and ordered their monsters to avoid Godzilla as much as possible, then both monsters would be free to cause destruction without interference.
Godzilla is extremely territorial. He also has a sort of sixth sense that lets him detect and home in on another monster. Even if the bad guys tried to leave him alone, Godzilla would track them down and destroy them anyways.
'Cause Japan is Godzilla's own 'hood, and ain't no-one trashin' his home turf!
They didn't want competition?
Also, in Final Wars, the Gotengo was directly leading Godzilla towards the monsters and the Xilian Mothership, even if X hadn't sent monster after monster after Godzilla, he'd ultimately have ended up having all his monsters fight him anyway. He was just being proactive and possibly buying time for Gorath to arrive with Monster X or for Gigan to be salvaged and repaired.
Except in both cases Godzilla would be unable to follow the monster in question. King Ghidorah can fly faster than Godzilla can swim or walk, making it easy to stay ahead of Godzilla while still allowing Ghidorah to freely destroy cities. In Final Wars, the monsters were held in stasis aboard ships until they were teleported to Godzilla's location. If they stayed aboard those ships then Godzilla would be unable to fight them.
Yes, but Godzilla would have then just gone straight to the Futurians/Xilians and attacked their ships. And remember that in both cases humans helped Godzilla by teleporting the Futurian ship to his location and leading Godzilla to the Xilian mothership.
Why do the humans keep sending the military after Godzilla and his kaiju co-stars throughout the Toho films, even when they've been shown to be completely useless against them? You'd think they'd've learned after the very first kaiju attack that they wouldn't be able to do jack shit to any of them. One guy in Godzilla Vs. King Ghidorah even admits that the humans are no match at all for Godzilla. So what's the deal?
I imagine that at least in some cases, its to keep them distracted so civilians can safely evacuate.
This is explicitly the case in at least a couple movies; they discover that the more aggressive kaiju will go after anything they regard as a legitimate threat, so they act as bait to lure Godzilla and company away from evacuation zones. Plus, with newer kaiju, they might as well give the military a shot, just in case they actually can hurt them (as happened to Gaira).
How is Japan (or any country for that matter) able to afford all the damage Godzilla and his colleagues cause throughout the franchise?
Insurance claims, raising taxes, quantitative easing. The usual things governments do when they need to magic up extra cash from nowhere. Money only exists as far as governments say it does, so it isn't really that much of a problem.
What you SHOULD be asking is how does Japan rebuild so quickly after kaiju attacks?
So, in The Return of Godzilla the cast used a specialized transmitter attract Godzilla. Given that this plan worked so well, why was it never brought up again in the Hesei films? It wasn't until King Ghidorah that Godzilla was mutated further into a 100 meter tall behemoth.
Well aside from the fact that it would defeat the purpose of keeping the big guy around for future films if you could do that, where in the hell would you actually lead him to? There's only so many volcanoes out there and Godzilla's not an idiot. He'd figure it out that the signal he was following was a little too obviously fake after the last time.
How are the humans able to determine a kaiju's origins so quickly throughout the series?
Multiple Kaiju, most notably King Ghidorah, have been said to have exterminated entire world more technologically advanced than earth. How? Even assuming highly advanced civilizations could never kill them (even though other Kaiju can simply beat them to death), they're very big and very slow, so hiding from them, underground or in the wilderness, and using radar and planes to run from them if you are discovered, seems trivially easy.
Earth has Godzilla (and Mothra in her 1990s Rebirth trilogy) protect itself from threats such as King Ghidorah. I'm guessing all those other planets weren't as fortunate.
Still doesn't explain how they wipe out advanced civilizations entirely, even if they can't be killed. The Anime needed to make Godzilla a serious Adaptational Badass for him to create a real threat of human extinction.