Extending semigroup theory?
The structure you suggest is known as a 'semihypergroup', a notion introduced by F. Marty in 1934. According to Google, some of your questions have already been attacked by a diverse community of mathematicians involving Greeks, Iranians and Thai people. The most challenging question seems point (iv) but I'm unsure about the outcome, though a positive result in that directions would probably have far-reaching consequences -- possibly in relation with Rhodes' interest in the separation of complexity classes.