Your question is based on misconceptions. Those forms are in my humble opinion not "reduced".
How far should the German speak accurately and what does the Non-German know from school?
There is such a thing as high german, for sure. The more important question is: what does the German know from school?
The non-german knows nothing, while the ideal German speaker knows everything and doesn't need justification, by definition. You'll agree that this is not realistic. Therefore the definition sucks. It is caused by and reinforces the home-advantage which is usually in favor of the teacher, regardless of the pupils.
As far as I know, gaeden variety teachers of a second languages at the entry level receive fairly little training in historical linguistics. However, we cannot treat the question in purely synchronic terms, because of the definitory problem: Who decides what High German means?
Traditional philology is fraud with 1. polemics 2. appeals to authoroty and 3. medieval etymology.
For example, -te in biste is enclitic. The parent language might have had OV sentence structure without overt objects, as in pro drop languages like Latin (> ti amo "love you"). It is quite likely that that's the origin of the pronoun, but it's probably not the only deciding factor. While I don't understand the theory in it's entirety, or at all, my understanding is that this is a matter of current research since the idea of linguistics as an empirical science is relatively young and Donald Ringe for example admits that these paradigms are well complicated.
- Therefore it is quite unliky that schools could teach it correctly. The polemics is in this case from me, to distract from my own ignorance.
For example, 3pl sind wir is reconstructed Proto-Germanic *izum wīz, with apocope in the first and rhotacizm in the second iz (also seen in 1Sg.past war, cp. English was), which I have taken from Wiktionary on the condition that they could have it wrong, which would still support my argument. Proto-Italic *(e?)somos > Latin sumus "we are" is good evidence for the notion though we'd need a lot more to make a case. If taken at face value, there is no need for recourse to Bavarian 3Pl mir "we" from buccalization next to nasals (which happen to be in the ending of 3pl weak verbs), though I'm sure that's precisy what you had in mind with "reduction".
- I won't argue that it cannot have been a kind of contraction in origin, because, as said, even specialists consider it difficult, but I wouldn't call that Bavarian, besides, me here in the north says it too—even if I think I mean sind wir.
I'm pretty sure that Einwand-frei is based on a folk etymology. I didn't know about Eimannfrei and thought the variant Einmannfrei must be the correct one. In any event, wenden is for burger-flippers whereas Ei is possiy from Proto-Germanic *aiwo, cp. Old Saxon and Old English ae "law" (a horse I'm gonna ride to death) and want, German wähnen, respectively mund as in Vormund, or Lat. reprimand, whathaveyou, making the phrase equivalent to Rechtsbegehren or something like it. That hary even matters. However, since the spelling has been fixed long ago, it is what it is and it almost makes sense. Nobody says Ich hät ein **Eimann, indicating that it is not a general phenomenon.
As to your other points
Often it's overlaid with the regional dialect.
True. The extend to which phonology can be incompatible is enormous. Language Development research shows the acquisition of native phoneme inventory is completed in infancy. That's all I know about that because linguistics is not on the common curriculum. So, I'm not sure if clean High German is the best option or even a reasonable goal in intrapersonal communication. Rather I'd argue personal experience has shown that speech will converge to a degree. That's an obvious fact with so-called Kiez-Deutsch, too, which is under researched and a hot-topic in current research to the point that there's no fixed name for the phenomenon—which concures not so much with the lreceding answer.
Then hammer can turn to hammor ...
which is in line with *u (see above), IMHO. The etymology of haben is diffult anyhow.
... or biste can turn to bischt.
That's not colloquial but dialectal. It has been adopted in certain phrases in Kiez Deutsch, though ("beschte Leben").
I understand it as appropriations to engl. gonna or wanna.
Different subject, doesn't matter. But see above wähnen.
Of course, etymology as the true meaning as well as "entimology" as the true sectioning is no magic bullet and entirely relative. The question is esse tially too broad because it questions what schools can and should do, which depends on your framework that may be highly opinionated (ie. I'd consoder beating a form of non-verbal communication that one should be able to understand but I won't encourage corporal punishment in educational institutions, and I cannot argue for segregation either without painting myself into a corner).