1 I agree, but if Spain conquers England, then England becomes a part of the Spanish Empire, so his English policies will probably also be less tolerant.
2 The also very catholic Charles V had a similar difficult relationship with the papacy.
At some point also the
Catholic Kings, Isabella and Ferdinand had a difficult relationship with the Papacy. The origin of the problem, I think, was on the role of the religion in the scheme of the Hispanic Monarchy. As a raising autoritary monarchy, the first hispanic monarchs saw the religion as a political tool or, if you want to say it differently, as an extension of their political power. That has much to do with the origin and longevity of the Spanish Inquisition, which in many senses was a political-ideological police, and also has to do with the ethnic cleasing against jews and muslims.Through it could seem favourable to the church's interests in fact it was a breaking up with the traditional relation between the Papacy and the christian princes, not unlike to Gallicanism,
mutatis mutandis. In that sense, we can say that the Reformation helped a bit the Hispanic Monarchy since it made the Papacy "somewhat" dependent of the spanish support and less likely to contest the throne's aspirations regarding church's affairs.
Regarding your point 1, I agree about the possible religious repression in England, but besides Philip II politics, would english catholics demand, or even lead, some action against the protestants?
Anyway, as others have said, I have doubts about the possibilities of Spain to retain England much time. It would put new economical and logistical problems and, if there is serious resistance, Philip's manpower could became overstreched.
Well, Philip II could take the throne as he did in Portugal, but Portugal is in the backyard of Spain and their nobility and institutions seem more likely to accept or at least tolerate Philip's rule than the english ones and perhaps Philip's claim over the portuguese throne was more defendible than the english one. Also, if Portugal can be used as example, Portuguese affairs reamined mostly in portuguese hands during Philip's reign, so an english inquisition is not inevitable.
If it happens as Tyr proppose, who could be the candidate/s to the english throne?
Cheers.