Monks and nuns aren't ordained; if a monk was ordained in the new rite (or more specifically, by a new bishop) then the doubt would extend to such a man.

I don't think the vows they've taken are canonical, though that isn't to say that they wouldn't be legitimate personal vows in some instances, when the candidate is truly earnest and pious.

Hello there, just joined.  I've read a lot about traditionalists' opinions on the validity (or lack thereof) of Novus Ordo Masses and ordinations of priests, but I'm [wondering] about the mendicant orders such as the Carmelites and Dominicans.  If one considers ordinations of priests invalid does this extend to monks and nuns?  (Apologies for my ignorance).

Don't forget the Servites and the Franciscans.

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There is no such thing as an "invalid" nun or monk.

However, joining or remaining in a Novus Ordo convent or monastery is a great danger to the faith and members of the order will not find salvation simply because they are in those orders if (or, more likely, when) they lose the faith.

This anecdote illustrates some of the problems with Novus Ordo religious communities:  I happen to know a woman who had entered into a Novus Ordo Carmelite convent.  She found that she was being forced to choose between Holy Obedience to superiors or committing mortal sin.  She told me that access to the sacrament of confession was limited but the superior required her to receive communion even if she was conscious of serious sin because it was scandalous for a sister not to receive communion.  This was just one issue she had grave problems with.  She was encouraged to depart the convent as the superiors felt she was a "trouble maker".  She is not a sedevacantist and does not consider the Novus Ordo priests to be invalid and attends the Novus Ordo even now.  Of course, this is an isolated case, but given what we hear from Novus Ordo priests and bishops on a daily basis, I doubt that problems such as these are truly limited.

The Conciliar sect is a danger to the Catholic Faith irregardless of the externals.  

(And don't bother telling me that "irregardless" is not a word.  It is a word, it conveys a meaning, and everyone who read what I wrote understood the meanings I intend.  In short, effective communication happened.  Only snobs complain about its use.)

Archbishop Lefebvre considered the ordinations of the Novus Ordo to be valid. Carmelites, Franciscans, and other monks and friars make a religious profession. God will judge the legitimacy of their profession based on how they live out that profession.