Learn Hardcore Russian: У моей соседки насморк, поэтому ей хочется сидеть дома и пить тёплый чай. - My female neighbor has a runny nose, so she feels like staying at home and drinking warm tea.

Russian often expresses possession or “having” with the pattern у + GENITIVE + (есть) + noun.
So У моей соседки насморк literally means At my (female) neighbor, (there is) a runny nose, i.e. My neighbor has a cold/runny nose. In present tense, есть is usually omitted.

What case is моей соседки, and why?

It’s genitive because у requires the genitive case.

соседка → соседки (genitive singular)
моя → моей (genitive feminine singular to match соседки)

Does соседки mean “neighbors” (plural) or “neighbor’s” (singular)?

Here it’s singular genitive: (of) my neighbor (female).
It looks like nominative plural (соседки = “(female) neighbors”), but context and the preposition у tell you it’s genitive singular.

What exactly does насморк mean? Is it “a cold”?

насморк specifically means a runny nose / nasal cold (nasal congestion/runny nose).
English often says “a cold,” but Russian distinguishes:

насморк = runny nose
простуда = a cold (more general)

Why is there a comma before поэтому?

Because поэтому here introduces a result clause (“therefore/so”), and Russian punctuation typically separates the cause and result with a comma:
У моей соседки насморк, поэтому... = My neighbor has a runny nose, so...

What does ей хочется mean grammatically? Why not она хочет?

ей хочется is an impersonal construction meaning she feels like / she has the desire to.

ей is dative (“to her”)
хочется is the impersonal form (“it is wanted/desired”)

Compared to она хочет (“she wants”), ей хочется often sounds more like a feeling/urge, sometimes softer or less deliberate.

Why is it ей (dative) and not она (nominative)?

Because with хочется the person experiencing the desire is expressed in the dative:

мне хочется = I feel like
тебе хочется = you feel like
ей хочется = she feels like

This is a common Russian pattern with feelings/states.

Why are there two infinitives: сидеть and пить?

After хочется, Russian uses the infinitive to say what someone feels like doing:
ей хочется сидеть... и пить... = she feels like sitting... and drinking...
The two actions are joined by и (“and”) and share the same хочется.

Why is it сидеть дома and not something like “to stay at home”?

Russian often uses сидеть дома literally “to sit at home” to mean to stay/keep to home, especially when you’re ill or resting. It’s a very natural phrase and doesn’t necessarily imply sitting in a chair the whole time.

What’s the role of тёплый in тёплый чай, and why that ending?

тёплый is an adjective meaning warm. It agrees with чай (masculine singular nominative/accusative in this context).

чай is masculine → тёплый (masc. singular)
Also, пить чай takes the direct object in the accusative, and for inanimate masculine nouns like чай, accusative = nominative in form, so чай stays чай.

What’s with the letter ё in тёплый? Can it be written as теплый?

Yes, ё is often written as е in everyday text: тёплый → теплый.
But the pronunciation is different: ё is pronounced yo ([ʲo]). When learning, it’s helpful to remember the correct ё even if many texts omit the dots.