Learn Hardcore Russian: Я горжусь сестрой, потому что она стала хорошей медсестрой. - I am proud of my sister because she became a good nurse.

Russian usually expresses “I am proud of X” with the verb гордиться (to be proud), not with the adjective гордый.
So the natural phrase is я горжусь сестрой (“I am proud of my sister”), literally “I-pride-myself-by-sister.”
You can say я гордый but that means “I am a proud (prideful) person” in character, not “I’m proud of someone right now.”

What is the ending -юсь in горжусь? Is this a reflexive verb?

Yes. Горжусь = горжу + -сь, and -сь / -ся is the reflexive ending.
The infinitive is гордиться (горди́ться). All its forms are reflexive: я горжусь, ты гордишься, он гордится etc.
Many Russian verbs that describe feelings or attitudes toward something are reflexive and take an object in a specific case (here: instrumental).

Why is it сестрой and not сестра or сестру?

Сестрой is the instrumental case of сестра.
The verb гордиться (“to be proud of”) always takes the instrumental case: гордиться кем? чем? (whom? what?).
So you must say я горжусь сестрой, я горжусь другом, мы гордимся ребёнком, etc., never accusative or nominative.

How does the noun сестра decline, and where does сестрой fit in?

Сестра is a regular feminine noun ending in -а. Its main singular forms are:

Nominative: сестра (who? what?) – “sister” as subject
Genitive: сестры (of whom?)
Dative: сестре (to whom?)
Accusative: сестру (whom?)
Instrumental: сестрой / сестрою (by/with whom?) – this is the form in the sentence
Prepositional: сестре (about whom?)

Here we use the instrumental because of горжусь.

Shouldn’t it be моей сестрой (“my sister”)? Why is there no мой?

Russian often omits possessive pronouns when the relationship is obvious, especially with close family members.
Я горжусь сестрой is naturally understood as “I am proud of my sister.”
You can add it for clarity or emphasis: я горжусь моей сестрой or more idiomatically я горжусь своей сестрой.

What does потому что mean exactly, and are there alternatives?

Потому что means “because” and introduces a reason: Я горжусь сестрой, потому что… (“I am proud of my sister because…”).
You could also use так как or поскольку, which are a bit more formal or “bookish”:

Я горжусь сестрой, так как она стала хорошей медсестрой.
But потому что is the most neutral, everyday choice.

Why is there a comma before потому что?

Russian normally separates clauses with a comma when they are joined by потому что.
We have two clauses:

Я горжусь сестрой
(потому что) она стала хорошей медсестрой
Because the second is a full clause giving the reason for the first, a comma is required: …сестрой, потому что…

Why do we use стала here? Why not just say она хорошая медсестра?

Стала is the past tense of стать – “to become.”
Она стала хорошей медсестрой means “she became a good nurse,” emphasizing the change and the achievement.
If you say она хорошая медсестра, that just states her current quality: “she is a good nurse,” without the idea of having become one.

Is стала perfective? What’s the imperfective, and what would change?

Yes, стала is the past of the perfective verb стать. The imperfective counterpart is становиться.

Она стала хорошей медсестрой. – She became and now is a good nurse (result).
Она становилась хорошей медсестрой. – She was in the process of becoming a good nurse (focus on the process, not just the final result).
In this sentence, we want the completed result, so стала is best.

Why is it хорошей медсестрой and not хорошая медсестра?

After verbs of becoming like стать, Russian normally uses the instrumental case for the new role or status: стать кем? чем?
So we need instrumental forms: хорошей медсестрой (feminine instrumental) rather than nominative хорошая медсестра.
Compare: Он стал врачом (“He became a doctor”), она стала учительницей (“she became a teacher”).

How do хорошей and медсестрой agree grammatically?

Both words are feminine singular in the instrumental case.

Nominative: хорошая медсестра
Instrumental: хорошей медсестрой
The adjective changes from -ая to -ей, and the noun from -а to -ой, to match feminine instrumental.

What does the word медсестра literally mean? Is it always feminine?

Медсестра literally comes from медицинская сестра – “medical sister.”
It is grammatically and conceptually feminine; it refers to a (female) nurse.
For a male nurse, Russian typically uses медбрат (“medical brother”).

Could the word order be different, like Я горжусь, потому что она стала хорошей медсестрой, сестрой?

That kind of word order would sound unnatural or confusing.
The standard and clear order is: Я горжусь сестрой, потому что она стала хорошей медсестрой.
Russian word order is flexible, but here splitting горжусь from сестрой or moving сестрой to the end would feel very marked or odd.

Where does the stress fall in горжусь, сестрой, and медсестрой?

горжусь – stress on -юсь: gor-ЖУСЬ (горжу́сь)
сестрой – stress on -ой: ses-ТРОЙ (сестро́й)
медсестрой – stress on the last syllable: med-ses-ТРОЙ (медсестро́й)

The stresses of сестрой and медсестрой match in the instrumental singular.

Can we drop the pronoun она and say …потому что стала хорошей медсестрой?

Yes, in context it would still be understood and is grammatically correct:
Я горжусь сестрой, потому что стала хорошей медсестрой.
However, including она is a bit clearer and slightly more neutral, especially in written or careful speech.