À l’université où ils étudient, il y a une grande faculté de lettres.

Both à and dans can translate as “at / in”, but they’re not interchangeable.
À l’université means “at university” in a general, institutional sense (the place where you study or work).
Dans l’université would mean physically inside the building and is much less natural here.
For institutions like à l’école, à l’université, à l’hôpital, French normally uses à, not dans.

Université is feminine, so the basic form is à la université, but French avoids having two vowels in a row.
When la comes before a word starting with a vowel or mute h, it becomes l’ (this is elision).
So à la université → à l’université.
The same happens in l’école, l’amie, l’hôtel, etc.

Où is the relative pronoun used for places and times, and it means “where” (or “when” for times).
In l’université où ils étudient, où refers back to l’université and means “where”.
Que and qui refer to people or things, not places in this way, so l’université que ils étudient would be incorrect here.
You could say l’université dans laquelle ils étudient, but où is shorter and more natural.

It can mean both.
French usually uses the simple present tense (ils étudient) where English might use either “they study” (habitual) or “they are studying” (right now).
Context tells you whether it’s about a general situation or something happening currently.

Il y a is a set expression meaning “there is / there are”.
The il here is a dummy subject (an empty pronoun), similar to “it” or “there” in English in sentences like “There is a problem.”
So il y a une grande faculté de lettres = “There is a large faculty of arts.”
You do not say “y a une grande faculté…” in standard written French, even though you might hear “y’a” in casual speech.

Yes, that word order is also correct and means essentially the same thing.
Starting with À l’université où ils étudient places emphasis on the location (“As for the university where they study…”).
Putting Il y a first is more neutral and straightforward.
Both are fine in normal French.

No, this is a classic false friend.
In French, une faculté is usually a division / department / school within a university (e.g. the faculty of law, faculty of science).
It refers to an institutional unit or building, not to the teaching staff as a group.
The sense “the professors” would be le corps enseignant, les professeurs, etc.

Lettres in this context means literature and the humanities, not alphabet letters.
Une faculté de lettres is roughly a humanities faculty, often focused on literature, languages, philosophy, history, etc.
Other examples: faculté de droit (law), faculté de médecine (medicine), faculté de sciences (science).

When one noun is used to qualify another in French (like “faculty of arts”), the pattern is often [noun] + de + [noun] without an article.
So we say une faculté de lettres, une école de musique, un professeur de mathématiques.
Using des (faculté des lettres) can appear in some fixed names or more formal titles, but faculté de lettres is the standard generic form.

Faculté is a feminine noun, so it takes the feminine article une, not un.
The adjective grand must agree in gender and number with the noun it modifies, so it becomes grande (adding -e for feminine singular).
Hence: une grande faculté.

Most adjectives do come after the noun, but a common group (often called BANGS/BAGS: beauty, age, number, goodness, size) usually comes before.
Grand / grande (size) is one of these, so une grande faculté is the normal order.
Other examples: un petit livre, un bon film, une vieille maison.

Où with an accent means “where” (or sometimes “when” in certain expressions of time).
Ou without an accent means “or”.
They are pronounced the same in modern French, so you have to distinguish them by context and spelling:

• Où ils étudient = “where they study”
• Ou ils étudient ou ils travaillent = “Either they study or they work.”