Learn Hardcore German: Meine Schwester ist auch allergisch, aber sie bleibt gelassen. - My sister is allergic too, but she stays calm.

Because meine Schwester is the subject (nominative case) of the first clause: Meine Schwester ist ….
You’d use meiner Schwester (dative) only when the sister is an indirect object or after certain prepositions, e.g. Ich helfe meiner Schwester. / bei meiner Schwester.

Why does the sentence use ist?

ist is the 3rd person singular present tense of sein (to be):

ich bin
du bist
er/sie/es ist
So with Meine Schwester (= she), you use ist.

What does auch mean here, and where can it go in the sentence?

Here auch means also/too. In this sentence it modifies the idea “allergic,” so placing it near allergisch makes that clear: ist auch allergisch.
German word order is flexible, and different positions shift emphasis:

Meine Schwester ist auch allergisch. = My sister is also allergic (in addition to someone else).
Auch meine Schwester ist allergisch. = Even my sister is allergic / My sister too (emphasis on “my sister”).
Meine Schwester ist allergisch auch… is generally not idiomatic.

Why is there a comma before aber?

Because aber connects two independent clauses (each could be a full sentence):
1) Meine Schwester ist auch allergisch
2) sie bleibt gelassen
In German, it’s standard to use a comma in this situation. (In English it’s often optional depending on style, but in German it’s commonly expected.)

What’s the role of aber compared to sondern?

aber means but/however and introduces a contrast without necessarily negating the first clause.
sondern is used after a negation and means “but rather/instead.” Example:

Sie ist allergisch, aber sie bleibt gelassen. (contrast)
Sie ist nicht krank, sondern allergisch. (correction after negation)

Why is it sie bleibt gelassen and not sie ist gelassen?

Both can exist, but they don’t mean exactly the same thing:

sie ist gelassen = she is calm (a state/description).
sie bleibt gelassen = she stays/remains calm (she keeps her composure despite something).
Using bleibt highlights that allergy doesn’t disturb her calmness.

What exactly does gelassen mean, and how is it different from ruhig?

gelassen means calm, composed, unflustered, often with the nuance of “taking things in stride.”
ruhig means quiet/calm and can refer to sound or general calmness, not necessarily composure under pressure.
So gelassen bleiben is a common collocation meaning “to keep calm / stay composed.”

Why is gelassen not changed for gender (like an adjective ending)?

Because gelassen is a predicate adjective after a verb (here bleiben), not an adjective directly before a noun. Predicate adjectives in German usually have no endings:

Sie bleibt gelassen. (no ending)
But before a noun, it does take endings:
eine gelassene Schwester (a calm sister)

Is allergisch a noun or an adjective, and why doesn’t it take an ending?

allergisch is an adjective. Here it’s also used as a predicate adjective after sein, so it doesn’t take endings:

Sie ist allergisch.
If it were before a noun, it would take endings:
eine allergische Reaktion (an allergic reaction)

Could I replace bleibt with wird?

Not with the same meaning.

bleiben = remain/stay (no change) → sie bleibt gelassen = she stays calm.
werden = become (a change happens) → sie wird gelassen = she becomes calm (she wasn’t calm before).

Why is sie repeated? Could it be omitted?

It can’t simply be omitted in standard German because the second clause needs a subject: sie bleibt gelassen.
You could avoid repetition by restructuring, e.g.:

Meine Schwester ist auch allergisch, bleibt aber gelassen.
Here Meine Schwester is still understood as the subject of bleibt, and aber is moved into the clause.

Is gelassen bleiben a fixed expression?

Yes, it’s very common and idiomatic. You’ll often see:

Bleib gelassen! = Stay calm!
In Stresssituationen gelassen bleiben. = To remain calm in stressful situations.

What’s the typical spoken emphasis/intonation for this sentence?

A natural pattern is:

Slight emphasis on the contrast: … allergisch, aber …
Emphasis often falls on gelassen, because it’s the surprising/contrasting part:
Meine Schwester ist auch allergisch, aber sie bleibt geLASSen.