With the note that the on-topic guidelines for this Stack Exchange don't include requests for proofreading, let's address some of the concerns you've already raised.
"Being a [X], Y is..." is grammatically valid but, yes, awkward. An improvement would be "As a private and ACO-accredited school, Valley East Side is...." Why? Perhaps ...
 I, [having played well], am proud of my game.
 I, [having been playing well], expect to win.
Preliminary point: the bracketed elements are not adjective phrases modifying the subject "I", but non-finite subordinate clauses.
The subordinate clauses are not modifiers, but supplementary adjuncts presenting non-integrated content, though they ...
The needed nomenclature can be chosen as that provided in §§ 3.54 and 3.56 from "A Comprehensive grammar of the English language"; and as well a system for naming all complex verb forms is given.
3.54 There are four basic types of construction in a complex verb phrase:
Type A (MODAL)
consists of a modal auxiliary + the base of a verb: eg: must