Can there ever be a functionally complete set of grammar rules which can parse any statement in English (locale-specific) accurately and which can be possibly implemented for use in AI-based projects?
Parse it yes, accurately most likely no.
According to my understanding on how we derive meaning from sounds, there are 2 complementary strategies:
A rule based system for ordering words to facilitate communication, here meaning is derived from interaction of discrete sounds and their independent meaning, so you could parse a sentence based on a rule book.
E.G. "This was a triumph" : the parser would extract a pronoun (This) with corresponding meaning ( a specific person or thing ) ; a verb (was) with corresponding meaning ( occurred ); ( a) and here we start with some parsing problems , what would the parser extract, a noun or an indefinite article ? An so we consult the grammar rule book, and settle for the meaning ( indefinite article any one of ), you have to parse the next word and refer to it though, but let's gloss over that for now, and finally (triumph) a noun ( it could also be a verb, but thanks to the grammar rule book we settled for a noun with meaning: ( victory,conquest), so in the end we have ( joining the meanings ):
A specific thing occurred of victory. Close enough and I am glossing over a few other rules, but that's not the point, the other strategy is:
A lexical dictionary (or lexicon)
Where words or sounds are associated with specific meaning. Here meaning is derived from one or more words or sounds as a unit. This introduces the problem to a parser, since well, it shouldn't parse anything.
E.G. "Non Plus Ultra" And so the AI parser would recognize that this phrase is not to be parsed and instead matched with meaning :
The highest point or culmination
Lexical units introduce another issue in that they themselves could be part of the first example, and so you end up with recursion.
if it is possible for a computer to parse a well-versed sentence written in English as if it were parsed by an adult English-speaking human?
I believe it could be possible, most examples I've seen deal effectively with the grammar rule book or the lexicon part, but I am not aware of a combination of both, but in terms of programming, it could happen.
Unfortunately even if you solve this problem, your AI would not really understand things in the strict sense, but rather present you with very elaborate synonyms, additionally context (as mentioned in the comments) plays a role into the grammar and lexicon strategies.
If it cannot be represented using simple grammar rules, what kind of semantic structure can be used to generalize it?
A mixed one where there are both grammar rules and a lexicon and both can change and be influenced based on the AI specific context and experience as well as a system for dealing with these objects could be one way.