Unfortunately, you really picked the wrong Sony platform for that sort of thing, at least right now. I'd really recommend a PS3 at the moment for those sort of things right now.
Still, as far as Metroid-style games go, I think there's a lot of Metroid in the level design of Bloodborne, which I was able to really get into despite not playing the previous Souls games. However, it is M-rated for good reason and you said you were trying to avoid M-rated experiences. Axiom Verge is another popular Metroid-style Indie game, though I didn't much care for it. I do, however, strongly recommend the Metroid/Castlevania-inspired Dust: An Elysian Tale. There's also the Strider remake, which I enjoyed despite a slow start.
When it comes to platformers, it's pretty barren on PS4 right now. LittleBigPlanet 3 is excellent and has the best SP campaign of the series, but it has a very different feel than Mario. The Last Tinker is decent and charming (and often on sale), but it's far from the most polished platformer. The Ratchet & Clank remake that's coming out in April, though, looks incredible and I'd recommend you pick that up once it releases. If you want to hate yourself, there's also Super Meat Boy, which some masochists out there really enjoy. Tearaway Unfolded best fits the platforming genre, but it's much more geared towards creation & exploration akin to a Zelda game. Tembo the Badass Elephant is nice, simple fun as well. Unravel is delightful and gorgeous (and surprisingly inexpensive), but it's a pretty rudimentary platformer.
As far as Zelda-style adventure games, I'd once again point to Bloodborne. However, if you want to dip into the PS2 games available on PS4, I have heard very good things about Dark Cloud 2 (I'm still trying to make my way through Dark Cloud 1, which is far from great). If you like your hacking and slashing on the cheesy, self-deprecating side, I'd proudly point to Atlus' The Deadly Tower of Monsters, a silly hack & slasher themed around "so bad it's good" B Movie Sci-Fi.
In any case, I hope I gave you some ideas of things to check out. Incidentally, if you want a fun Smash-esque competitive fighting party game, check out Towerfall: Ascension. Just be warned that it's local multiplayer only.