Yes, Wolverine's claws could pierce or lacerate Thor's flesh. If he can cut the Hulk with those claws, (and he can), Wolverine would certainly be able to injure Thor with them. That doesn't mean that Wolverine would defeat Thor in a fight, however.
Mjolnir also has the advantage of being enhanced by magic and holds the power of a cosmic storm within it. In the case of Thor's Mjolnir versus Wolverine's adamantium claws, Thor's hammer has a clear edge.
Wolverine was mind controlled into perceiving Thor as Sabertooth, but still, it's not an alternate reality or a What If type story, so it's a representative fight. Wolverine was able to cut Thor and make him bleed, which angered the big guy and lead him to end the fight quickly with lightning and hammer shots.
If we are playing by the canon rules of the comic Marvel Universe from Earth-616, without the Odin-force or other fundamental reality-alteration taking place, Wolverine's adamantium claws should completely resist Mjolnir 's attack. No. Mjolnir is protected by magic.
His adamantium claws are one of the few weapons in the Marvel Universe that can consistently pierce Hulk's skin and Wolverine has no problem going for Hulk's head. In fact, if it came down to it, Wolverine could cut Hulk's head off, ending the Hulk once and for all.
Wolverine vs Blob - "Did You Just Call Me Blob" Scene | X-Men Origins Wolverine (2009) Movie Clip 4K
Can Wolverine cut vibranium?
Because without altering any continuities, if Fox's Wolverine and Marvel's Captain America were to come to blows, the claws would slice through the shield like a hot knife through butter. Vibranium can absorb kinetic energy, but it isn't indestructible (as evidenced by Thanos destroying the shield in Endgame).
Wolverine, given enough time, could defeat Colossus in a fight using only his claws, but he would have a terrifically difficult time doing so; the latter's skin is made of adamantium, same as Logan's claws, and even a demon-controlled Wolverine can do very little against him.
There have been times where Magneto has sort of wielded the hammer in the comics. In the Ultimate universe, Magneto gets a hold of the hammer by manipulating the electromagnetic field around the hammer, thus using it but not technically wielding/holding it, circumventing any need to be worthy.
It is NOT enchanted. Most people make this mistake but it was confirmed by the directors that anyone who possesses the ''physical strength'' to lift the weapon can lift it. How was Thanos able to lift both Mjölnir and Stormbreaker? He was able to lift Stormbreaker because Stormbreaker is purely about strength.
Magneto can manipulate adamantium. Magneto has always been able to affect adamantium. This happens in many of different stories, including when Magneto slowly pulls Wolverine's adamantium-coated skeleton straight out of his body.
Magneto rips the Adamantium from Wolverine's bones, killing Wolverine, leaving just a severely charred skeleton and an arm of flesh due to his healing factor not working.
Even if the fight happens with the stones, it'll very difficult for Thanos to kill Wolverine but eventually, Thanos will win given that he has the most powerful weapon in the world. But if the battle is between Thanos without the gauntlet and Wolverine, then Wolverine will definitely defeat him.
At the end of the day, Wolverine could defeat Captain America if they were pitted against one another on multiple occasions. Sure, Cap has the will to keep going even when at the end of his rope, but so does Logan. Both are outrageously stubborn and forceful.
With only four exceptions, Wolverine would never be able to cut through Superman skin, with his adamantium claws. Those exceptions being: Superman was mostly drained of his stored energy. Superman was exposed to red solar energy.
The general consensus was that Wolverine's claws could theoretically puncture Superman's skin, but that it would require tremendous super strength behind them to do so, which Wolverine doesn't have.
Stormbreaker has the additional power to summon the Bifröst, allowing Thor to teleport anywhere in the Nine Realms. Screen Rant has identified Stormbreaker as the more powerful of the two, based on this ability, as well as it being a much larger and edged weapon.
He traveled to Nidavellir and forged Stormbreaker, a weapon not only strong enough to kill Thanos but counter the Infinity Gauntlet and even summon the Bifrost. However, the weapon wasn't ready until Groot completed it by dismembering himself - and, in doing so, he became the first person to lift Stormbreaker.
Stormbreaker can be used by anyone, it isn't bound by Odin's enchantments to only be lifted by the worthy. Mjolnir is, however, still enchanted, meaning that only those that are worthy may wield it. This applies to your last two questions as well. If you can lift Mjolnir, you can wield it.
However, Thanos is still standing, and he forces Odin to unleash the full might of his power, the Odinforce. In an epic sequence, Thanos keeps advancing toward his foe, withstanding blast after blast of deadly energy. In the end, Odin is too strong, and Thanos is clearly beaten.
If Marvel fans have ever wondered if Spider-Man was worthy of lifting Thor's hammer, the answer is yes. Just NOT the Spider-Man you think… Although he didn't really meet Thor beyond a couple shared scenes in Avenger's Endgame, Spider-Man has shown he's actually a big fan of the Thunder God.
Although Cap's internal conflicts prevented him from being fully worthy to lift Mjölnir during Age of Ultron, Cap really almost pulled off stealing Thor's thunder.
Wolverine's vaunted claws can cut the Hulk's skin, but they can't do diddly-squat to the Juggernaut. Wolverine can't hurt the Juggernaut, but the Juggernaut sure as Hell can hurt Wolverine.
So while Juggernaut is much more powerful, and would normally be able to win in a fight, Colossus could convince Cyttorak to transfer his power, or even cut off Cain Marko's connection to Cyttorak. Cain Marko would be left as a normal human. Colossus would still be a powerful mutant.
Wolverine isn't just armed with enhanced senses and an impressive healing factor - his skeleton is also encased completely in adamantium. His claws are sheathed in the metal, allowing them to be razor-sharp and could certainly cut through Iron Man's armor.
