Is there any historical record of Elizabeth the First's reaction to John Knox's tract on 'the monstruous regimen'?

Simon Schama wrote ‘A History of Britain’ (BBC 2000) and I quote from Vol. 1, page 342 as to when, back in the day, Tyndale’s 1526 English Bible first referred to women as “the weaker vessel” (1 Peter 3:7), and many Protestant manuals on proper household regimen routinely quoted it and exhorted the subservience of women to men. The kingdoms of Britain were viewed as

“a household writ large. To John Knox. The Scottish Calvinist preacher, Marian exile and author of The First Blast of the Trumpet against the Monstrous Regiment of Women (1558), the reign of so many women – Mary Tudor, Mary of Guise in Scotland, Catherine de’ Medici in France – was a ‘monstriferous abomination’, a species of plague. It was, in fact, the obvious explanation why the times were so manifestly out of joint. Women, Knox wrote, echoing the commonplace of the day, ‘ought to be constant, stable, prudent and doing everything with discretion and reason, virtues which women cannot have in equality with men… Nature I say doth paint [women] further to be weak, frail, impatient, feeble and foolish and experience hath declared them to be inconstant, variable and cruel and lacking the spirit of counsel and regiment… in the nature of all women lurketh such vices as in good governors are not tolerable’. Learning of Elizabeth’s accession, Knox was concerned enough not to damage the chances for a Protestant government in England to write to William Cecil, the queen’s new secretary of state, and explain (though not recant) his diatribe. Like many other critics of female rule, Knox was prepared to concede that Elizabeth might be considered a special case, sent by God to fulfil his purposes of restoring the gospel. But his insistence that she must nevertheless acknowledge that womanly rule was ‘repugnant’ to the Almighty’s proper order was not calculated to endear him to the young queen.”

Knox was not the only one subjected to Elizabeth’s threat of landing on English shores at risk of death; she had earlier denied her cousin, Mary, safe conduct through England, forcing her to sail the long rout offshore to Scotland (p353).

One source for more detailed information on Elizabeth’s attitude to Knox (post-Monstrous rant) is the book John Knox by Lord Eustace Percy (Lutterworth Press reprinted it in 2013). One of his remote ancestors features marginally in the Knox story. His book was first published in 1937 but does not claim original research and there are no footnotes. Percy is apparently both a discriminating admirer and defender of Knox. Percy quotes frequently from Knox’s sermons, letters, addresses and the history of the Reformation which, according to Randolph puts ‘more life in us than five hundred trumpets continually blustering in our ears,’ (Taken from a review of Percy’s book by Donald Mackay of Edinburgh, June 2014).

However, another book of the same title, 'John Knox', but by Jane Dawson promises the academic research you seek for this matter. It was published in 2015 by New Haven and London: Yale University Press. Professor Dawson is the joint chronicler of extensive correspondence between Knox and Christopher Goodman, who was a divinity professor at Oxford before he was exiled in Frankfurt during the reign of Mary Tudor. He was Knox’s closest friend for most of his life. Professor Dawson’s PhD was on the subject of Goodman. There is significant new material on Knox’s theory of resistance to tyrannical rulers. [It may answer your point 1 question.] Dawson also delineates the catastrophic impact on Elizabeth I of England of Knox’s infamous ‘First Blast’. Dawson points out that this work of Knox’s may probably be attributed in large part to the failure of English Puritanism to become embedded more firmly in Anglican church life, certainly during her reign. After its publication, Knox was only ever to enter England in great secrecy, such was the Queen’s hostility. [In response to your point 2 question I would think Elizabeth did not so much hinder Knox in his Christian ministry as compel him to focus it far more on Scotland and Europe than on England.]

I have taken that last paragraph largely from a review of Dawson’s book written by Professor of Church History at Edinburgh Theological Seminary, John McIntosh. As I do not have either of the books I recommend, I trust this inadequate answer will at least send you on your way rejoicing, resolved to get your hands on them!