I started gathering as much information about the Mythos as I could, and I'm attempting to put it into some sort of novel. I'm making it a frame narrative story that shifts from protagonist to protagonist, as they each have some sort of adventure dealing with Lovecraftian horrors. Would this type of story work, or should I change it up a bit?
Depends on what you want to do with it. If you plan to publish the story you write you can't plagiarize, so no.
I was reading some Cthulu Lore and related stuff in wikipedia etc, and I think there might be copyright issues if you use the actual mythos itself as someone has ownership of it. I'm no legal expert mind you, but you might want to look into it further. You may have to twist it into your own unique interpretation or something? Much like you can't write about a Hobbit in a land called Middle-Earth, but you can have a short statured hero going on an adventure with wizards in Yee-Olde-Earth.
As far as copyright goes, I have to admit I don't know for sure, but I have read books explicitly using Cthulhu mythos that were not by Lovecraft (and some that were written very recently). You may have to pay for that right or something, or it may be public domain, I don't know. As far as the story idea, it could work but it depends on how you do it. A huge part of the mythos is the philosophy that no human can (or really should) know about how the universe really works. Using your framework, you could do some pretty fun stuff. I think it'd be neat to see something like all the different characters getting small pieces of what's going on, and then the reader having to put those pieces together like a puzzle. Or, to way overextend the puzzle analogy, have it missing a piece, just to leave them guessing. Maybe even have characters reference an unseen person who would have that last piece, but, in true Lovecraft fashion, went insane because of their knowledge? That's my take anyways. As I said, there are a lot of directions, but in general it's a good framework. Lots of options for what you can do with it.
I know for a fact that you can use the mythos. No one owns copyright on the ideas, and though Arkham House claims copyright on a lot of the literature, most of this claim is utterly groundless.
copyrights I looked into the actual copyrights of Lovecraft's works, and it seems to be up for public domain, so I shouldn't have much issues there. In the actual story, I was going to base it off of an organization that knows about the Great old ones, Outer Gods, etc., and tries to fight them off, and each chapter would be based off an individual mission of them trying to destroy or seal one of these monsters. Since it's non-traditional Lovecraftian horror, would people still read it?
I am a big fan of Lovecraft and I've love to read this. As for the copyright issue, it's crazier than an outhouse rat. All of Lovecraft's stuff is public domain, along with the extended stuff written by his peers. Much of what he wrote was based on stuff that itself was public domain at the time. (Cosmic horror is an old, old concept.) You have to be very selective and make sure you're using his material - or that of his peers - and not anything written more recently. Particularly anything belonging to Chaosium. Very easy if you are doing your research by reading his stories and not searching the net for Cthulthu-related material. If you're going the traditional route, no worries. Arkham House never threatens to sue anyone with money.
I believe that lovecraft actually wanted other writers to use the cthulhu mythos and there is pretty much a whole cthulhu genre of books.
Yeah, many of Lovecraft's author friends would work his mythology into their stories. One of the more notable examples is Robert E Howard, who wrote the original Conan the Barbarian stories. Some of the eldritch abominations of the mythos are actually present within the Conan universe. Not sure if he ever wrote a story where Conan fights Cthulhu or his worshipers though...
What? He/she can't write a fanfic version without plagiarizing? I don't see why not. Maybe you see something I don't?
Cthulhu him/itself? No. But Conan messed up plenty of eldritch-worshipping cults in his day. Though such things were toned down to better fit the setting. Such creatures always had to take some earthly form to interact with our world. So even the worst of them could be killed with fire or sword, as opposed to many of Lovecraft's creatures that were not always comprised of matter as humans understood the term. Though one of the stories (The Devil in Iron) deals with Conan fighting a god that has chosen an iron statue as its form on this earth. Good luck hacking that to bits with a sword.