And a healthy debate to boot! 
One anime that might be worth a look I recall from when I was a kid is a girl's baseball team - "Princess 9" I think it's called? It's much shorter than Prince of Tennis.
Most Ghibli films are a safe bet. But be careful. Some of them like Nausicaa do have violence and killing.
Most Ghilbli films are pretty awesome but they do feature guns or other issues: Porco Russo (guns also a very brutal fist fight), Howl's moving Castle (guns also a lo of violence, some very slight male nudity), Princess Mononoke is hands down my favorite anime movie of all time but waaaay too violent for a school setting also features ancient guns, Princess Kaguya is going to come down to one issue the princess unknowingly tries to tempt her childhood friend to run away with her and he hides the fact he's married with children from her and nearly does.
Wolf Children we discussed earlier has some major issues despite it being mostly a beautiful film.
In any case, @Rodimus13 , I think your best bet for getting insight on all these on short notice is.. watching trailers and maybe even reading a bunch of reviews for all these. You can find the age ratings on MyAnimeList by looking up each anime to see if these are appropriate enough, though it won't tell you what exactly gives it that rating, but it's better than nothing. If all else fails, watching these anime and reviewing them yourself would give you better results as far as seeing how appropriate they'd be. I'd definitely recommend a 3 episode rule for this, seeing how many shows were recommended here already~
Bomberman jetterz was cute and bittersweet, but it has bomberman (the game character) so... maybe not
Wolf children tho! Damn was that movie beautiful but prolly not for 13 yos.
Summer Wars was great tho! and iirc had no nudity/sex mentions.
Princess Tutu, but the problem is ahiru turns naked when she gets hit with water while a duck. It's not sexualized imo, but it happens, moreso for comedy if anything else.
Also!! Letter to Momo!!!!!! It was an amazing movie that I loved to death and is great for kids.
Space patrol luluco, but it's rated pg-14 rip
natsume's book of frーah, pg-14 again, fuck
napping princess is pg, but I haven't seen it yet. Will get bsck to this
I wouldn't say no to spirited away as that ties to brothels is sooo obscure that I doubt even most adults in the US would pick up much less for kids. We have a much higher probability of some one connecting it to satanism than that. But if we worry about that then anything that's doesn't come from the bible or has conflict would be off.
Just so so you know if you show an anime of any kind you will get one person complaining even in something as innocent as pokemon. Speaking of which Monster rancher is a good anime for kids similar to obviously digimon and pokemon.
Edit: I wouldn't recommend spice and wolf naked Holo for most of the first episode and some other episodes and lots of talks about church (as well as critiques). That anime is more for a 15+ audience
How about Natsume Yuujinchou? As far as I remember there is no sexual jokes on that anime. And most of episodes have beautiful plot. And chubby, old cat
Oh man I have no clue what would be PG-13... yeah i'd be best if you watched or at least read the episode summaries first 
Second Hamtaro, Magical DoReMi, Chii's Sweet Home, and K-ON.
Yumeiro Patissiere, Barakamon?
Cells at Work is educational, but it's kinda bloody when WBC goes to eradicate the foreign bodies
I saw someone mention Piano no Mori, I never watched it but the MC does live in a brothel and was almost sexually assaulted (in the manga).
Oh my bad... I'm bewildered because my comics I'm making actually features 3rd year middle school students, but they are 16-17 years old and forgot most of them are 12 to 14 years old 
Also I'm kinda surprised that it's pg14. Maybe because that cat is an alcoholic
You can't go wrong with Little Witch Academia -- It's like a PG Harry Potter style atmosphere but in an all girls school with an all boys school next door. Super pretty, it's funny, good for all ages, no random sexy stuff you'd feel weird about. Also, it takes place at a school and you're watching it at a school.
That's actually the main problem - most anime are off the list for middle schoolers, even high schoolers, because of their content. It's less about what the teachers think the kids are okay watching, or what the kids want to watch, and more about what the PTA will get in an uproar over. The younger the kids, the more presence of the PTA.
And actually, I was thinking of the Jesus and Buddah anime from the weird manga- I haven't seen it but something tells me even that wouldn't be acceptable in a school setting.
mostly not as mostly it will be seen as the wrong version. "kids should be reading the true king james edition of the bible they will love that".
It will be alot easier if it was what will be a good children anime rather than a good anime that will not get the PTA's undies in a tizzle.
Doremon is a preschooler's anime, I haven't seen it but it seems to be mostly about this robot cat solving a bunch of kids problems. That sounds harmless and it has a bunch of episodes so you can pick and choose which episodes without worrying about long running plot plot
Watch you some Digimon specifically season 3 its not only the best season but also will hit you hard in the feels in quite a few places
When I was in middle school I was a huge fan of Ojamajo doremi and the pretty cure series(still on going).
They're demographic is around 7-12, but Japanese kids shows tend to be a little more mature than American kids shows. The English dubs of those shows (Magical DoReMi and Glitter Force), are way too heavily edited to the point that it dumbs down the original story and characters.
But you can check out both and gauge which you feel is more appropriate
2 years later
I think you can watch Death Notebook if you don't know, it's an anime about a young prodigy who found one day that very Death Note, with the help of which he started to kill different criminals. It's a very interesting anime about psychology, I think if you watch it, you'll have something to discuss at school later. I can also advise you to take a quiz on your stereotype, or what kind of anime hero you are, I think you would be interested.
If it wasn't for the "no guns," I'd recommend Girls und Panzer, which is utterly insane and amazingly fun (and teaches just about every life lesson I would ever want my daughter to learn) - and my 4 year old little girl loves it.
Otherwise, Bofuri is a LOT of fun.
Astro Boy (2003) (Might be a problem with the no guns rule)
San-Gatsu no Lion (It has the best arc about dealing with school bullying, has a heavy arc in the first season)
Cells at Work (Regular one, Black! is a bit darker)
Bakuman (Anime about making manga)
Shirobako (Anime about making anime)
SSSS.Gridman (A fun giant robot anime)
Keep your hands off Eizouken (Anime about making anime: Electric Boogaloo Auteur edition)
I would recommend Sakura Card Captor, but I remembered all the kinda weird couple dynamics it has. Revolutionary Girl Uthena is another kinda iffy choice, depends what is your goal with the anime club.
@ar-ninetysix
Your lie in April? You want to traumatize some kids?
Christ, this was typed out a few years ago lol anyways-
I'm sticking with it being in my recommendations list! Granted, I have not seen it in forever, but from what I do remember.. it deals with the stresses of being a prodigy, the sacrificing time spent on friendships for practicing a hobby/ potential career, parental abuse, and to some extent grief over the loss of a loved one.
You might be thinking, why on Earth would I recommend it then if it has all these themes? I truly believe it would be a disservice to try to shelter teens from media that showcases all of that. I can't pretend to know that teens are old enough to understand the concepts, but I think there's room for discussion afterwards.
We don't know what they dealing with at home, and similarly, we don't know what they're gonna deal with once they're out of school. I just figured if there was a way that they had to be exposed to that stuff first, it'd be better for it to be through a show that could still be enjoyed a fair amount, you know?
But for the sake of this reply! I do kiiinda agree. Put an asterisk on Your Lie in April if you feel so inclined, OP