cowboy bebop....
no schoolgirls
no talking flying animals
no yelling the name of the move a person is about to do in combat.

Prison School ( ͡° ͜ʖ ͡°)

I also only really like it in the original language not dubbed and watching the shape of voice, right now :)

Trigun :D

Since you mentioned mostly films, I'm just throwing a hilarious one here:
Tokyo Godfathers

I'm not into anime that much, but you mentioned two Miyazaki movies, have you seen others? I liked almost every movie I have seen from Miyazaki or Studio Ghibli. Here are some you may want to look up: Princess Mononoke, Ponyo, Whisper of the Heart, The Secret World of Arrietty, Nausicaa of the Valley of the Wind, Howl's Moving Castle, Castle in the Sky, When Marnie Was There, Kiki's Delivery Service.

I have not seen these yet, but plan to: The Wind Rises, Porco Rosso, Pom Poko.

A couple other movies I can remember that I liked are Your Name and Ghost in the Shell. Also, I love Little Nemo, but I grew up with it and have a lot of nostalgia, I'm not sure what other people think of the movie. You can actually watch it on youtube here.

I think I have only watched one anime series and I really enjoyed it, it was Elfen Lied. You probably won't want to watch this one with your family around.

Not a weeb but ...

• Demon Slayer: Kimetsu no Yaiba
• Fire Force
• Samurai Champloo
• Psycho-Pass
• Attack on Titan
• My Hero Academia
• Given
• The Rising of the Shield Hero
• Yuri!!! on ICE
• K (a.k.a K Project)
• Tower of God
• Sword Art Online

Naruto and One Piece are great shows but there is a shit ton of fillers and are in general huge time sinks. One Piece is really bad with this and it is currently moving at a snail's pace. Still great shows, but worth keeping this in mind.

The only anime I've seen is Grave of the Fireflies. I enjoyed it.

I have seen a lot of anime but i shall give u 3 suggestions

1. Ghost Hound
2. Akira
3. Devils line

Anything by Masaaki Yuasa, including but not limited to:

1. Mind Game
2. Kaiba
3. Walk On Girl
4. Devilman Crybaby

Good ol' sci-fi thrillers:

1. Texhnolyze
2. Paprika
3. Serial Experiments Lain
4. Psycho Pass
5. Ghost in the Shell

Shit that doesn't really fit your requirements, but you should watch anyways:

1. Steins;Gate
2. March Comes in Like a Lion
3. Id: Invaded

For a good chuckle:

1. Ghost Stories (dub)
2. Garzey's Wing (dub)

Rurôni Kenshin (1996–1999)

Violet Evergarden (series 1st, then movie)
Macross franchise

JoJo's Bizarre Adventure!!! The manga started in the 80's but the anime adaptation is fairly recent. There was a 90's OVA though, but I'd recommend the TV anime as it follow the manga a lot more closely

One Punch man, it is a super cool anime!

I really loved Slayers. The ovas were kinda meh, though.

I'm not that into anime but Jojo's Bizarre Adventure is really fun.

Dorohedoro is on Netflix, it's a bit chaotic but really worth it after you just stop trying to understand what's going on.

My favorite anime/animated series are The Tatami Galaxy, Steins;Gate and Serial Experiments Lain.

Magic Knight Rayearth - a fantastic story with great plot twists
009-1 - a short anime with a narrative that puts blockbuster shounens and pseudo-intellectual animes to shame

Some of these have been mentioned but I'll bring in my opinions on some I've enjoyed. Now, I don't watch very many but the ones I do are the ones I've stuck with. I'm glad you have some rules as I have some general rules for committing to an anime too, namely:
a) Less than 100 episodes. 2 seasons are usually best because it gives the story time to develop and reach a conclusion. The longer a show goes on (even non-anime) the more weird shit the writers have to do to keep interest and plots moving, so forget it.
b) No "anime faces" (there's one exception in the list). This is probably similar to your no cringe rule, I just don't need a character to suddenly pull a very cartoony angry face to tell me they're-angry-but-it's-funny while a sound affect goes whoop whoop bee doop. It's the reason I sadly couldn't even get started on Full Metal Alchemist.
c) The less fanservice the better. Especially characters constantly tripping and falling onto someone's crotch/cleavage/buttcrack. If this happens in S1E1, I'm out.

With that in mind, here are some that I thoroughly enjoyed.

Attack on Titan - I burned out on this awhile back because its later seasons set up a lot of conflict that was invisible earlier and so there's been much less actual attacks on titan. Still, I got back into it for the batshit crazy answers the later seasons delivered and would still recommend it, especially for the intense first season at least.

Steins Gate - time travel shenanigans. Very entertaining with lots of memorable lines/characters. I think someone falls onto someone's chest at one point, but at least it's late in the series and it's more of an attempt at a light moment when the show is at its darkest.

Neon Genesis Evangelion - I tried watching the remake movies, but there's something about the original's shoestring budget charm and real sense of vision, as wild as they wanted. Much less polished than current Boardroom Decision Targetted Demographic-Friendly Entertainment Product. I see echoes of Evangelion in Attack on Titan as well.

Fate Stay Night / Fate Zero - Fate Zero is the superior series, but you'll get more out of it if you watch Night as Zero is its prequel. Fate Stay Night does drift into tropes on occasion but it's still an entertaining premise which keeps things moving. I've lost track of the series after Zero, every now and then I see some new Fate Avergablathally Bang Bang title on Netflix, I'm like wth is this. Some are new stories I think, some are retellings of the original with different endings, but I don't have time to make sense of it all. I really doubt any of them can top Zero anyway, so why bother.

One Punch Man - very much the anti-anime, it takes great pleasure in poking fun at lots of tournament/overpowered hero type shows. Kind of manages to have its cake and eat it too. Loads of fun to watch.

Cowboy Bebop - I'm told this is very good but somehow it never clicked. Need to try again as I like the setting. Killer theme though, regardless.

Code Geass: Lelouch of the Thing - I think they remade this too, I mean the original series in this case. I guess I just like mecha anime. I think the blinder you go into this one, the better. Good twists and a mix of supernatural and big robots.

Death Note - half of this series is really worth your time. I didn't see all of it but I read the manga and when Something Big Happened it just lost steam in the second half. Still, the tension and cat-mouse interplay between killer/detective is thrilling and the premise on its own is very original.

Fight! Iczer One, Iczer Reborn AKA Adventure!! Iczer-3 and Iczer Girl Iczelion are my three l time favorites.

(not so) fun fact- I first encountered it with Iczer-3 at a video store what had a 3 for 2 days for 3 dollars rental dead. I watched the first part, RAN back to rent the second part and saw it was a cliffhanger. NOBODY had the third tape. I called every place I had a membership to and even several I didn't (this was before the internet, mind you) and nobody had it. Fortunately, I knew of a comic book store in my town that's STILL in business to this day that would import it for me. So I imported all three parts and fell in LOVE with it.

Also, Grave of the Fireflies was one of the most AMAZING anime movies I have STILL seen to date. That and Voices of a Distant Star. That one I downloaded it illegally. After I watched it, I HAD to run to the local Best Buy to own the dvd. Voices has one of the single most moving OST I have heard in a long time. Hard to believe the whole thing was done by ONE person.

Here are some series I enjoyed that haven't been mentioned yet.

Astra Lost in Space
Dr. Stone
Gabriel Dropout
Girls' Last Tour
Keep Your Hands Off Eizouken!
Laid-Back Camp
Non Non Biyori
Somali and the Forest Spirit

Well, recently im watching Detective Conan (Case-Closed) and it´s that type of mind-game anime, it´s one of the best anime´s i´ve ever seen. It have 900+ episodes and 24 movies