From what religious perspective is Supernatural aproached? I've seen god, Lucifer, darkness (Amara), angels, demons... pretty much all levels from heaven to hell. Is Jesus ever mentioned? If he does exist in that world, how come he hasn't showed up to clean daddy's mess? .
To answer your questions more or less in order.
From what religious perspective is Supernatural approached?
Simultaneously none of them and all of them. Confused? I'll explain.
The monsters that the Winchesters and other hunters face cover a wide spectrum from the Native American Wendigo to the Japanese Shojo. Interestingly, most of the monsters they face have weaknesses based on the culture they hail from. For example, Wendigo was held back by Anasazi symbols while Shojo could only be killed by a sword blessed by a Shinto ritual.
There doesn't seem to be a one size fits all kill method (not counting the Colt) or a faith/culture that is supreme over the others (though some creatures are naturally stronger and I'm getting to that).
We also see that Pagan gods exist from many pantheons Hindu, Canaanite, Norse, Celtic and Greek to name a few.They are not simply dismissed as "monsters" posing as something more than they are but are recognized as gods. Lucifer is able to kill some of them easily but that only speaks to how much stronger he is.
Is Jesus ever mentioned?
Yes, and it seems that Christian rituals have just as much power over Christian monsters as rituals from other cultures have over their monsters. For example, in Houses of the Holy, Sam uses a Christian seance to summon the spirit of Father Gregory. Also, before the boys learned the Enochian Exorcism, they used a version of Psalm 68 from the Christian Old Testament.
But that isn't directly mentioning Jesus
You're not wrong. So let's look at some quotes that put Jesus into perspective.
Rick: This emptiness you feel inside -- it's really a hunger for him.
Eve: You do know that Jesus was just a man.
This is important because Eve was stated to be older than the Angels, meaning she was alive when Jesus walked the Earth and presumably knew more about Him than a trucker somewhere in the Midwest.
From The Future
Dagon: You get the chance Mary of Nazareth, Part 2, Evil Jesus Edition.
Here, we have one of the most powerful demons in existence, apparently, recognizing the power that Jesus has/had and is seemingly happy that "this time" that power will be used for evil.
If he does exist in that world, how come he hasn't showed up to clean daddy's mess?
There are a couple of options here all of them are based on the fact that God/Chuck didn't fix the problems.
Option 1: Chuck is Jesus
In the Christian Trinity, The Father, The Son and the Holy Spirit are all God but are separate from one another.
Chuck is identified as God but not specifically as God the Father. It is heavily implied that Chuck is God the Father in that the angels all refer to him as such but not stated outright
The entire Trinity existed BEFORE the first angel was made. In fact, the First Council of Nicea found
that the Son was true God, coeternal with the Father
This all leads to the possibility that ANY Person of the Trinity would be recognized by the angels as their Father.
In Exodus 33:19-20, God (presumably the Father) said that nobody could see His face and live. However, when Jesus (God the Son) was born as a human, nobody died when they looked at him even when he was transfigured.
- The accounts of the Transfiguration in the New Testament describe Jesus becoming a bright light. Similarly, in Don't call me Shurley, Chuck reveals who he is by becoming a bright light.
So if Chuck is Jesus, this explains pretty well why he didn't clean up the mess. It was what he wanted.
Option 2: Chuck is not Jesus
In this option, Chuck may very well be God the Father or, somehow, the entire Trinity represented as one guy (blasphemy!). In this case, you're essentially asking why God (the Son) has not come to Earth to undo what God (the Father) has done, fix what God (the Father) left broken etc.
Presumably, God has his reasons and no Person of the Trinity will act against the will of another because it is their shared will as God.
What about Joshua?
In spite of some fairly significant circumstantial evidence that Joshua is meant to be Jesus, I feel like that's a bit of a stretch considering that Joshua was killed by (an admittedly very powerful) demon indicating that he was just an angel.
It should be noted that whether or not this answer is correct depends on the Christian/Catholic Church evolving the same way in the Supernatural reality as it did in ours