I actually think that a lot really depends on the monster Link is fighting. Th Zelda series like most fantasy worlds in fiction are large enough to easily contain many various forms of wildlife, while some are just animals that are from there like raccoons or sparrows are on Earth, others could be various chimera creatures created by ambitious or foolish sorcerers. Even demonic entities summoned by those same sorcerers, to monsters crafted by the gods as servants or for vengeance, magic and divine intervention could play a huge part in just where most of the normal monsters came from. Quite a few monsters in the games really are just various forms of wildlife that occurs naturally in the world of Hyrule. Things like Deku Babas, Octorocks, even chus could even be a regular animal based on how common they are in the series.
Moblins, Bulblins, and Lizalfos are commonly thought to be just other tribes that live in or around Hyrule just as Gorons and Zoras. Only they are not aligned with the Hylian society at all and are considered enemies to be defeated when they are encountered. Other more intelligent monsters have been considered to be normal Hylians that were transformed into monsters after joining Ganondorf's army in the Dark World. This was somewhat the explanation for moblins in ALTTP as they only appeared in the dark world and knights were the minions in the light world. But moblins have been appearing in the regular light world in most sequels since. Other more strange monsters could still be the servants of Ganondorf transformed by the power of the dark world.
Gibdos, Stalfos, and other kinds of undead appear to be something that happens independent of Ganondorf's interference. They were present in Hyrule before he gained the triforce, and many were around in Termina as well. Hyrule has a lot of restless dead including poes, which are sometimes imps in the world and other times are ghosts that cannot rest at all so they attack the living. Sometimes they are cursed by the land like the stalfos and skull kids created by the Lost Woods, there might be other areas where people are transformed by the power of an area like that. The Lost Woods of FSA transformed all those who enter into Deku Scrubs.
I am sure the exploding into black mist or poofing away to leave a rupee behind is just something that the developers left there to maintain the E rating in the games. Zelda could not really keep an E rating if Link really did cleave into everything he found leaving a trail of bodies throughout every dungeon. So the cartoon poof is still there today.
