I've noticed that phrase structure rules have been very inconsistent over my studies. I've seen NP = (det)(adj)N ; NP = (det)N(PP); these definitions seem to change with context. Is it just because there's no strict phrase formalisms in language? Or am I missing something?
There are many ways to describe a given language, even in a given formalism. I would think that at any time, a language is always inconsistent. Diachronic evolution seems a good reason to believe that. Latin has slowly evolved into Italian, Spanish, Catalan, Roumanian, Portuguese and French (what did I forget?). Actually there have been many more dialects and variants that did not all survive. And the evolution has been progressive. How can you expect to have consistency in such an unstable context?
I do not know of such experiments, but I would guess that no two people will agree everywhere on the grammaticality of some sets of sentences.
But that shows most for the visible (surface) syntax of languages. One task of linguistics is to identify deeper and more stable concepts that structure the language in a more consistent way.
What matters for surface syntax, that you are considering, is more to understand adequate and tractable formalisms that will let you describe the language (almost) adequately, and develop techniques to attain from that the deeper structure. Distinct surface descriptions in the chosen formalism can hopefully give you similar understanding.
Actually, even the choice of the formalism can be open, and there are many competing. Even if you consider only phrase structure, the variant you have been using seem to be the context-free variant of Chomsky. But others, with interesting descriptive properties have been developped, such as tree-adjoining-grammars, to give an example. Then these formalisms can form a skeleton to be completed by more detailed information (features).
Also relations with the lexicon have to be established, and structural information and features can be attached to the lexicon. Interestingly, when choosing or designing a formalism, independently of the language to be described, there will be tension between the "phrase structure skeleton" and the lexicon, regarding where some of the structural information should be placed. Emphasizing the lexicon can also be done by lexicalizing the phrase structure skeleton (which is what Greibach normal form does for context-free languages).
And there are other very different kinds of formalisms, such as categorial grammars.
What matters is to acquire the knowledge, understanding and effective usage of natural language description tools. Then you use these tools to give the best description you can of the natural language structure. We do not all have to agree on what might be the best description of the structure of a natural language (hence the variations you complain about), but we need common descriptive tools to understand each other when discussing these issues. The formalisms are (part of) the language needed by linguists to exchange and discuss meaningfully about the natural language.