There's a lot to write about here and a lot of it is outside my expertise, but I shall try and answer some of your questions.

Elizabethan schools were not a state initiative (although many were started by people close to the Queen) and had no overarching coordination, so standardisation would have been difficult. There was, however, broad agreement about what should be taught: a (renaissance) Humanist curriculum that supported civic and social objectives. Latin was taught, but particularly Cicero (for his insights on government), Greek and possibly Hebrew, character-building exercise, logic (usually via Aristotle). The exact details of this, and what it meant, are a subject of historiographical debate. We do know that during the Stuart period there was a backlash claiming too little English literature was taught. Near the end of Henry VIII's reign, a royal committee produced English and Latin grammar books and forbade the use of other texts (the combined edition is often referred to as "Lily's Latin Grammar", or the Royal Grammar) and Elizabeth's government prescribed its continued use in 1559... but how much they enforced this, we don't know. There's a reign-by-reign narrative of the development of schools in Derek Gillard's Education in England, and I've tried to summarise the most relevant points here.

There is quite a substantial grammar school "boom" during Elizabeth's reign. 136 schools are founded between 1558 and 1603, and another 83 1603-1625 (in the context of there being only 400 schools in England at the beginning of the reign of Henry VIII). The exact type, survival, and patronage of schools can be a bit ambiguous in this period. We have a much better picture of universities (see Lawrence Stone, "The Educational Revolution in England 1560-1640", 1964, here).

Legally teachers required approval by a bishop, under a 1559 statute. This usually meant they were Anglican clergymen, as they were required to subscribe to the 39 Articles and royal supremacy. I'm fairly sure there are records of prosecutions for unlicensed teaching (usually anti-Catholic in nature). There were inspections and examinations (for teachers) but these were haphazard. Elizabeth's minister Roger Ascham wrote an extensive guide to teaching (c.1570) but it's more concerned with suggesting appropriate content than worrying about methods. A typical Ascham comment:

With this way, of good understanding the matter, plain construing, diligent parsing, daily translating, cheerful admonishing, and heedful amending of faults: never leaving behind just praise for well doing. I would have the Scholar brought up withall, til he had read, & translated over the first book of Epistles chosen out by Sturmius, with a good piece of a comedy of Terence also.

"Sturmius" here probably refers to Johannes Sturm. Ascham has already, at this point, written several pages on the merits of Cicero.

In context, it was widely assumed that any clergyman would be able to teach (it was part of the training and duties). At the end of Henry VIII's reign Thomas Cranmer regarded the universities as the right source of educated clergy, which Ascham argued against - but it's clear that Ascham regards "teaching" as learned men setting an example, rather than a specific practice that can be learned. There's a discussion of clergymen's education up to the reign of Mary (and beyond) here.

In all this, you can see both continuity and a break with the past. The old medieval Trivium has been rejected, there is an interest in a standardised humanist curriculum (but little power to enforce it), but the thinkers/drivers of this all cut their teeth during the earlier Reformation. Like education throughout history, it's an ideological battleground, so it's hard to say whether teaching was good or bad without the filter of the religious and nationalist conflicts of the period.