What scientific conditions would need to be met for Santa to exist or at least for some things to work?
Santa and his abilities may very well exist outside of our universe. If there are indeed infinite universes, and the physics in each one varies ever so slightly... once you get far enough removed from our own universe, with its own laws of physics, the possibilities are theoretically infinite (the word "universe" itself could also be considered misapplied here)

pic huffingtonpost
Choosing the least fanciful question
How does he know who has been bad or nice?
There very well could be universes where the mental powers of the inhabitants are such that they're able to "know" a person upon sight or over distance. Able to communicate desires and information effortlessly and without limits. Perhaps in a not so radically different universe than ours only a few have "evolved" in such a way mentally.
Actually the other questions can also be addressed
How does Rudolph's nose glow?
Perhaps the nose of an animal glows through bioluminescence in one particular universe. Perhaps different body parts on other creatures also glow?
How does he fly?
Fly? Well, you'd have to get very far removed from our own laws of physics for this to be realistic. But hey, isn't that what infinity is best at? Getting us far removed?
So
Santa Claus? He is scientifically possible! See the sources I've provided below. One problem with the theory of chaotic inflation is that "everything" is moving away at such speed... there is no theory yet supporting the idea of catching another "reality".
I personally think that all my possibilities actually DO exist in reality, somewhere. And that's an edifying thought.
Happy Holidays
Sources:
Here theoretical physicist Andrei Linde explains Chaotic Inflation and the possible affects on our known physics - Universes with different physical constants
and because I enjoy Neil deGrasse Tyson I'll have to include some Neil too...
Cosmic Queries: The Multiverse with Neil deGrasse Tyson | Full Episode
here is a link to the challenge by @suesa
https://steemit.com/science/@suesa/suesa-s-science-challenge-3-a-very-science-christmas