In both the manga and anime, Tatsuya had a precious gift from his mother. In the anime, it was referred to as a toy phone. In the manga, it was a functioning walkie talkie (referred to as transceiver in the English translation). There just happened to be someone on the beach with another transceiver who was listening on the right channel. Asako's call ...
I guess you could describe this as calligraphic. I don't know enough to say whether this is what a calligrapher would consider genuine art vs. just being writing with bold and swishy brush strokes.
This writing doesn't "symbolize" anything, though - it's just the lyrics of the first verse of the OP theme as they're being sung.
混ぜんな mazen na
My theory about its "real name" is, either Towako, the name of hakumen's avatar, maybe Hakumen named its avatar the real name for self-satisfaction? and the second probable name, is kirio/Inasa, as in Kirio Inasa, the boy whom it kidnapped. after the kidnapping, it probably change his name as it likes.
I personally think, Towako is the "Real name".
Well, there definitely is an official soundtrack for Ushio to Tora. You can purchase one from here
As for your battle scene music, someone recreated the music on YouTube that you might find interesting.