I am about to apply to grad school as well, and I don't know any better than you do. So, the following are just my opinions:1.Is it alright to mention Professors names whom have helped you/ you look up to / you worked with in your SOP?
Yes, especially, if you have done some research work under their supervision. Also, if the prompt asks you to tell about someone who inspired you, then you have to name them.
2.Should I state what University I am coming from in the statement or does that matter?
Not necessary if you are including your CV.
3. Is it alright to use major specific jargon? (i.e. nihonjinron, sankinkotai, samurai, daibutsu, etc)
Yep. The person reading it will be quite familiar with the subject and its terms.
4. Should I assume that the reader knows about the major I am applying for and therefore not need to explain certain things (like the jargon above)? Or should I write like they know what I am saying?
Same as above.
5. Should I sign the botton of it?
Yes.
By the way, I like your username :D