Mi amiga quiere apuntarse al seminario de español del próximo semestre.

Because mi is a possessive adjective placed directly before a noun: mi amiga = my friend.

Mía is a possessive pronoun/adjective form used in different structures, for example:

• La amiga es mía = The friend is mine
• Una amiga mía = a friend of mine

So before the noun, you say mi amiga, not mía amiga.

No. The short possessive forms mi, tu, su do not change for gender. They only change for number:

• mi amiga
• mi amigo
• mis amigas
• mis amigos

So mi works with both masculine and feminine singular nouns.

Quiere is the third person singular of querer (to want):

• yo quiero
• tú quieres
• él/ella quiere

Here it means she wants because the subject is mi amiga.

In Spanish, querer + infinitive is a very common structure:

• quiere apuntarse = wants to sign up / wants to enrol

This works much like English want to + verb.

Because apuntarse is a reflexive verb with a different meaning from plain apuntar.

Some meanings:

• apuntar can mean to point, to write down, or to note
• apuntarse often means to sign up, to join, or to enrol

So in this sentence:

• quiere apuntarse al seminario = she wants to sign up for the seminar

The -se is important because it is part of the verb’s meaning here.

With a conjugated verb plus an infinitive, Spanish often allows two positions for reflexive pronouns:

• Mi amiga quiere apuntarse al seminario
• Mi amiga se quiere apuntar al seminario

Both are correct.

In your sentence, the pronoun is attached to the infinitive: apuntar + se = apuntarse. This is very common and natural.

In Spain, apuntarse is very common in everyday speech for signing up for something:

• apuntarse a un curso
• apuntarse al gimnasio
• apuntarse a una actividad

It is often a bit more informal or conversational than inscribirse.

• apuntarse = everyday sign up / join
• inscribirse = more formal register / enrol

In this sentence, apuntarse sounds very natural in Spain.

Al is the contraction of a + el:

• a el seminario → al seminario

This contraction is required in normal Spanish when a is followed by el.

So:

• apuntarse al seminario = to sign up for the seminar

A similar contraction appears later in the sentence with del.

Because apuntarse normally takes a before the thing you are joining or signing up for:

• apuntarse a un curso
• apuntarse a una excursión
• apuntarse al seminario

This is just the standard pattern of the verb. English uses for in sign up for, but Spanish uses a.

Del is the contraction of de + el:

• de el próximo semestre → del próximo semestre

So:

• del próximo semestre = of the next semester / for next semester

Just like al, this contraction is standard and required.

Because Spanish usually links nouns with de where English often uses a noun directly as an adjective.

So:

• seminario de español = Spanish seminar / seminar on Spanish

This pattern is very common:

• clase de música = music class
• libro de historia = history book
• profesor de inglés = English teacher

Using de español makes it clear that the seminar is about the Spanish language.

Because after de, when you are naming a subject or field of study, Spanish often does not use an article.

So these are natural:

• clase de español
• profesor de matemáticas
• seminario de historia

Here, español means the subject/language Spanish, so no article is needed.

Both positions are possible in Spanish, but they can sound slightly different.

• el próximo semestre is the most neutral and common way to say next semester
• el semestre próximo is also possible, but it can sound a bit more formal or stylistically marked

In everyday use, el próximo semestre is the more expected order.

Here próximo means next or upcoming:

• el próximo semestre = next semester

Depending on context, próximo can also mean near or close, but in time expressions like this it usually means next/upcoming.

Because seminario is a masculine noun in Spanish, so it takes el:

• el seminario

Its ending -o is a strong clue that it is masculine, though noun gender must ultimately be learned with each word.

That is why you get:

• al seminario = a + el seminario

Yes, but the tone and context change a bit.

• Mi amiga quiere inscribirse al/en el seminario...
  This sounds more formal: wants to register for the seminar

• Mi amiga quiere matricularse...
  This is more often used for officially enrolling in a course, programme, or school subject, not always for a one-off seminar

For a normal conversational sentence in Spain, quiere apuntarse al seminario is very natural.

Yes. Amiga is feminine, so it refers to a female friend.

If the friend were male, you would say:

• Mi amigo quiere apuntarse al seminario de español del próximo semestre.

Spanish nouns often show gender in endings like:

• amigo = male friend
• amiga = female friend

Yes, although the original order is the most neutral.

For example:

• Mi amiga quiere apuntarse al seminario de español del próximo semestre.
• Al seminario de español del próximo semestre, mi amiga quiere apuntarse.

The second version is more marked and might be used for emphasis. In normal conversation, the original word order is the most natural.