Mario has two types. The Zone Speed is the slow time one, while others like the Stopwatch are total timestops. There's also apparently a Battle Card with timestop abilities, but there's so many of those I can't quite find it.
Sleep Manipulation comes from the Lullaby Star Power. Can't find a video showcase, but here's a
screenshot. Very simple musical ability that puts opponents to sleep. There's Also Sleepy Sheep, which is shown in this
video
That Video Also shows the Capture Card right afterwards, which is the only Soul Manip I know for sure Mario has, the card's description even says "A card that catches an enemy's soul" which...yeah, that's a creepy description. Doubles as 'Sealing' too..So I was wrong about Mario not having Sealing, whoops.
As for invulribilities, Mario's got a bunch of ways. The classic Super Stars, of course, along with all the other 'invincible' power ups that make Mario almost entirely imprevious, though a few situations do negate the invincible power-ups such as falling off the stage so...take that for whatever level of Game Mechanics you will. Vanish Cap makes him intangible when he's wearing it. Gaurd Shell DX nullifies damage. Even if he does get Oneshotted he has the 1-Up Glove that gives him a passive resurection.
And ontop of explaining those, he still does have a lot of really great abilities otherwise. Durability negating attacks, 'insta-kill' moves, BFRs, Transumutation into gold, Probability Manipulations that make his opponents sometimes just straight up miss, stat changes, Mario's just got a tone he can do in Base thanks to having all sorts of really wacky gear.
However, now that Rainbow Dash and the MLP gang are in the Kilofoe range, that's a hefty AP advantage over Mario's 10 Foe (If I'm finally getting Base Mario's AP right...), so my question becomes how many of these Hax are negated by an AP differance? A few durability negating ones probably shouldn't, and are deadly combined with timestop, so Mario would still have a chance. ESPECIALLY Bloodlusted where he's not going to mess around and just go for his best options, so if enough of his Hax are still valid Mario might be able to completely ignore MLP's new massive AP advantage.