Learn Hardcore Spanish (Spain): Mi amigo universitario sigue estudiando español conmigo en la biblioteca. - My college friend keeps studying Spanish with me in the library.

Mi amigo universitario literally means “my university friend,” i.e., a friend who is a university student. Universitario works as an adjective describing him.
Mi amigo de la universidad means “my friend from university,” i.e., someone you met at university, whether or not he is still a student.

In many contexts they overlap, but amigo de la universidad is often more natural when you mean “the friend I met at university,” not “a person who is a uni student.”

Why is universitario after amigo? Could it be mi universitario amigo?

In Spanish, most descriptive adjectives normally come after the noun:

mi amigo universitario = my university friend

Putting it before (mi universitario amigo) is either wrong or sounds poetically marked/very unusual. So you should keep it as amigo universitario in standard speech.

What tense and person is sigue, and what verb is it from?

Sigue is:

from the verb seguir (to continue, to follow)
3rd person singular, present indicative: él / ella / usted sigue

It agrees with mi amigo universitario (he), so: “my university friend continues / keeps …”

Why do we say sigue estudiando and not just estudia?

Both are grammatically correct, but they don’t mean exactly the same:

estudia español conmigo = “he studies Spanish with me” (a general habit or fact)
sigue estudiando español conmigo = “he keeps / continues studying Spanish with me” (emphasizes that the action goes on, hasn’t stopped)

Seguir + gerundio is used to express continuity: sigue trabajando, sigo viviendo allí, etc.

What is estudiando? Is it like an English gerund?

Estudiando is the gerund (gerundio) of estudiar.

Formation: stem + -ando / -iendo

estudiar → estudiando
comer → comiendo
vivir → viviendo

It’s used here with seguir (seguir + gerundio) to express “to keep doing something.” It’s similar to English “studying,” but in Spanish you can’t use the gerund as a noun the way English does (“Studying is important” = Estudiar es importante, not Estudiando es importante).

Could I say continúa estudiando español instead of sigue estudiando español?

Yes, continúa estudiando español is also correct and means almost the same thing: “he continues studying Spanish.”

Differences:

seguir + gerundio is more common and slightly more colloquial.
continuar + gerundio is also frequent but can sound a bit more formal or neutral.

In everyday speech in Spain, sigue estudiando is very natural.

Why is it español and not el español here?

When talking about a language as an object you are learning or speaking, Spanish often drops the article:

Estudio español. = I study Spanish.
Hablo inglés. = I speak English.

You can say el español in some contexts (e.g. talking about the language as an abstract system: El español es una lengua románica.), but with estudiar un idioma, the more typical form is estudiar español, without el.

Why is it conmigo instead of con mí?

Spanish has special forms with con:

con + mí → conmigo (with me)
con + ti → contigo (with you, singular informal)
con + sí → consigo (with himself/herself/themselves, in some contexts)

So con mí is incorrect; you must say conmigo.

Can I change the order of conmigo and en la biblioteca? For example: … en la biblioteca conmigo?

Yes, it’s grammatically possible to change the order:

… sigue estudiando español conmigo en la biblioteca. (more neutral)
… sigue estudiando español en la biblioteca conmigo.

Both are understandable. The first is more natural because Spanish tends to put “with whom” (conmigo) before “where” (en la biblioteca), but word order is flexible and can change slightly for emphasis.

Why is it en la biblioteca and not just en biblioteca?

Most common place nouns in Spanish use the definite article when you refer to them in a specific, general sense:

en la biblioteca (in the library)
en el museo (in the museum)

There are a few set expressions without article, like en casa, en clase, en cama, but biblioteca is not one of these. So en biblioteca sounds wrong; you need la.

If my friend is a woman, how does the sentence change?

You need to change the noun and adjective to the feminine:

Mi amiga universitaria sigue estudiando español conmigo en la biblioteca.

Changes:

amigo → amiga
universitario → universitaria

Everything else stays the same.

Does sigue here mean “follows” or “continues”? How do I know?

Seguir can mean both “to follow” and “to continue/keep (doing something)”.

You know which meaning is intended by the structure:

seguir + a alguien → to follow someone (physically or on social media)
- Sigue a su profesor. = He follows his teacher.
seguir + gerundio → to keep doing something
- Sigue estudiando. = He keeps studying.

In your sentence, sigue + estudiando clearly means “keeps studying.”

Could I just say Mi amigo universitario estudia español conmigo en la biblioteca? Would that sound wrong?

It’s not wrong at all; it’s perfectly correct.

estudia español conmigo = he studies Spanish with me (neutral statement of fact)
sigue estudiando español conmigo = he is still studying / he keeps studying (emphasizes continuity)

Choose sigue estudiando if you want to stress that he hasn’t stopped or is still doing it.