¿Cuánto te cuesta la matrícula este semestre?

Te is an indirect object pronoun meaning to you / for you.
In Spanish, costar often works like: Something costs (to someone) X.
So structurally it’s: [La matrícula] (subject) + [te] (to you) + [cuesta] (costs).

Because the grammatical subject is la matrícula (singular), so the verb agrees with it: la matrícula cuesta.
You’d use cuestan if the subject were plural, e.g. Las matrículas cuestan mucho.

The verb is costar (to cost). It’s a stem-changing verb: o → ue in most present tense forms:

• cuesta, cuestan
• cuesto, cuestas, cuestan (etc.)

With an accent (cuánto) it’s interrogative/exclamatory: how much / how many.
Without an accent (cuanto) it’s usually relative: as much as / whatever amount.
Examples:

• ¿Cuánto cuesta? (question)
• Te pago cuanto me pidas. (whatever amount you ask)

In Spain, la matrícula commonly means enrolment/registration fee and often by extension the tuition payment for a term. Context (university, school) makes it clear.
It can also mean license plate (also common in Spain), but not in this context.

Spanish often uses the definite article (el/la/los/las) in places where English doesn’t. La matrícula is natural here because it refers to the enrolment/tuition fee (for that term).
You can omit it in some contexts, but it can sound less specific or more like a general concept: ¿Cuánto cuesta matrícula…? is much less standard than ¿Cuánto cuesta la matrícula…?

Both are correct. Adding te highlights that it’s your cost (what you personally have to pay), which can matter if different students pay different amounts (scholarships, residency status, etc.).

• ¿Cuánto cuesta la matrícula este semestre? = general price
• ¿Cuánto te cuesta…? = your price

Change te to le:
¿Cuánto le cuesta la matrícula este semestre?
(And if you want to be extra clear, you can add a usted: ¿Cuánto le cuesta a usted…?)

Yes, that alternative is completely natural. Spanish allows flexibility, especially with pronouns. Common options include:

• ¿Cuánto te cuesta la matrícula este semestre?
• ¿Cuánto cuesta la matrícula este semestre?
  The pronoun te typically goes before the conjugated verb (te cuesta).

It’s understood and correct. In Spain, you’ll also often hear este cuatrimestre (many universities divide the year into two cuatrimestres rather than “semesters” in everyday admin language). Usage depends on the institution.

Sometimes, but there’s a nuance. Valer is more like to be worth / to cost (price tag) and is common in casual price questions: ¿Cuánto vale?
For fees and what you pay, costar is very common and often sounds more appropriate: ¿Cuánto cuesta la matrícula…?