How's the cooperation between the US and Mexico anyway?
It's pretty good; the US and Mexican governments are good allies on good terms. This leads into the second part of your question...
Wouldn't it be cheaper to invade Mexico for a while, root out the drug trade, build up a less criminal economy, that kinda thing?
Absolutely not.
Not only does the Mexican constitution
expressly forbid foreign military intervention on their soil (even if via bilateral agreement), even if America were to say 'fuck that', it would be
very bad politically to
invade our southern friendly, non NATO, non NAFTA signatory neighbor to try to intervene militarily in something as fluid as the drug trade. It would be a horrible, terrible idea with literally no upsides.
Keep in mind that the cartels, specifically the Zetas are sitting on about 100 billion dollars of revenue, and they're extremely organized, with many of them being ex-special forces (the Los Zetas have a chain of command that includes
generals). The cartels aren't a bunch of shady thugs huddling around fires at night, they operate all the way into the echelons of the Mexican Army. Only the Mexican Navy is considered more or less not corrupt.
Could the U.S. military beat the drug cartels in a fight? Absolutely, but the cartels already fight asymmetrically, kidnapping families, putting people's heads on stakes, burning people alive, etc etc, and while American tourists are kidnapped and such, the cartels have no ideological interest in 'death to Americans' or anything, they just want to make money.
Now if that sentiment were changed via foreign invasion? The Mexican cartels are ludicrously brutal and have absolutely no hesitation to save their own profits/skins at the cost of anyone or anything. Yeah. You get the idea. Military wise, the cartels would stand absolutely no chance against a military crackdown, but the cartels don't fight like that against a superior enemy, and they wouldn't. They would fight in other ways, and those ways are generally pretty fucking terrible and things that all parties involved would like to be contained in the worst parts of Mexico.
The drug trade isn't something to 'root out' via force. The only real way to put a dent to it is the same way we got rid of the Mafia after Prohibition, end the war on drugs and crack down on corruption, the first one is on the U.S., and the second thing is endemic to Mexico, so that's on them.