Argthtjtr

I know that the noun faculty can be both singular and plural, but I’m confused whether it’s countable or uncountable when used in sentences like this one:

Many international faculty have difficulties with. . . .

It may help to think of "faculty" as both a collective noun and countable noun. In its guise as a collective noun, it may govern either a singular or plural verb, depending, respectively, on whether you're thinking of the faculty as a single group or as multiple individuals. The former is the default in AmE; the latter, in BrE. So in AmE, if I say

The faculty are revolting.

I mean that the members of the faculty disgust me. On the other hand, if I say

The faculty is revolting.

I mean that the faculty is about to storm the Dean's office.

In its guise as a countable noun, "faculty" may take its plural form "faculties" to indicate separate, multiple groups (and of course will then take a plural verb):

The three faculties of the School of Law, the School of Medicine, and
the College of Arts and Sciences are meeting in plenary session.

If you mean faculty as in the individual teachers within the group, then its common to say " faculty members" which is countable.

Otherwise faculty refers to the entire group, which is also countable.

So in your example:

Many international faculties (as in different groups)

Many international faculty members (individual members within the group)

But what about this, which I see often at work (in a university):

"Twenty faculty will be chosen to participate in the program... "

When used in sentences like that one, the use of the word "many" implies that "faculty" is being treated as a countable plural noun.

What other answers haven't said directly is that this is not the usual usage of faculty. Faculty is usually used as a singular count noun (with the plural form "faculties"), or as a plural non-count noun (through "notional agreement").

"Many international faculty" is not linguistically acceptable to me. Some speakers might have different intuitions, but deadrat and J. Paczuski's answers seem to be based on the same judgement of the unacceptability of usages like "many faculty". The answer by cestjill (with the example "twenty faculty") provides evidence for speakers who might not share this judgement.