You're a pokefarm user too, yay! Uh... anyway...
Is this a problem? Am I doing something wrong by writing these stories?
I assume you're asking this in a legal sense, and it's not a problem if you're not making profit from it. Make sure you've got disclaimers and all that good stuff. If' you're not asking in a legal sense, then: You can write whatever you want and post it (pretty much) anywhere, although obviously some places will just boot you for posting your story on their websites. But writing fanfiction isn't really a problem in this sense either.
Additionally, if I decide to publish my stories elsewhere than online (ie. physical copies)... is that even allowed?
Tch, publishing... I'm not particularly knowledgeable on Pokemon, but some authors expressed that you can't write fictions in their universes, some authors would prefer if you didn't profit from their material, and so on. Also, some places are strict about copyright infringement and so on. In Japan, publishers couldn't give less of a crap about people publishing works from their copyrighted material, though. That's why doujinshis are rampant over there. Rampant. In Japan, publishers think of doujinshis as pretty much "free press coverage" so... they don't care. In America (which is where I am, I don't really have a frame of reference for outside of America, sorry), that's grounds to get you sued. What you see in manga is, very likely, a result of better selling doujinshi (or just doujinshi artists being asked to make official material), but... the culture in Japan and the cultures outside of Japan towards copyright is way, way too different for me to tell you it's cool for you to publish your work in America (or elsewhere) without seeking permission first. When you ask about publishing a fan work (physically) on fanfiction websites, you normally get rerouted to answers like this. You can distribute your work not-for-profit here in America electronically, but I'm not sure about publishing a hard copy and distributing that. It's okay if it's copies for you and your friends, but you seem to want to publish and sell, like a business? In which case, no, not in America, you can't do that.
That being said... You're allowed to sell physical copies of your fan work at conventions. You still need to know the stance of the copyright holder (and there are more licensing issues if you're outside of Japan), because they can absolutely still sue you into next week. But conventions normally set limits as to how much you can sell (for example, only 300 copies of your self-published comic or something like that, though the pricing is also kind of important) and they also tend to have "permission" from licensing companies, who send lawyers to check out the works being sold. If your work does have mature themes, you will probably have to notify convention staff. They'll put you in an artist alley section (if they have it) for mature audiences. That way, younger people can't buy your material.
Doesn't mean you can go crazy (you should check the limitations to your publishing rights) but you absolutely can sell your fan work at an Artist Alley of your choice. Just know you're not going to be making billions off your work (or thousands. You'd be lucky to pull like $300 with one work alone, depending on your convention). Lawyers patrol the Artist Alleys, so you can get called out if they think you're overstepping your boundaries. You sound like you're a writer, someone who isn't producing a comic or artwork, I mean. I think they might read your stuff at the convention or at least page through it? Definitely talk with the convention staff if you want to know what to expect from the lawyers at conventions. Also, talk to people on websites outside of Pokefarm. DeviantArt is probably your best bet for finding out about Pokemon licensing initially (there are a lot of people who do this sort of stuff on DeviantArt, so it's a good place to ask about where to look for information, who to ask for information, what things are okay, and what things aren't okay) but if you want to be safe, branch out from there. It's a legal issue if this is the route you're pursuing, so I can't help and I don't know how much Writers SE can help.
But like I said you can still absolutely be sued. But you'll get a Cease and Desist notice first, most likely. Parodies normally get off with a little more lee-way, fan-art and fan-work is subjective. Again, this gets into legal issues I don't know much about, so...
In general, you need to really ask around about your particular fandom before publishing works for profit from that fandom willy-nilly. If you want to sell your work, you have to do it at the right places-- a comic book store can't sell your work next to licensed material, a bookstore won't take your fan novel, etc. etc. Long story short, do your homework!