Da meine Schwester heute früh losmuss, macht sie sich nur ein schnelles Rührei.

Not here. In this sentence, da is a subordinating conjunction meaning since, as, or because.

So:

• Da meine Schwester heute früh losmuss = Since my sister has to leave early today

This is different from da meaning there.

A useful nuance:

• da often sounds like the reason is background information or something the speaker treats as already understandable.
• weil could often be used instead, but da is especially common at the beginning of a sentence like this.

Good question. This is because losmüssen behaves like a separable verb in this kind of structure.

Compare:

• Main clause: Sie muss heute früh los.
• Subordinate clause: ..., da sie heute früh losmuss.

So in a main clause, the parts are split:

• muss ... los

But in a subordinate clause, the finite verb goes to the end, and the separated part joins it:

• losmuss

This is very similar to other separable verbs:

• Er fängt an.
• ..., weil er anfängt.

So losmuss is normal here.

Yes, in everyday German it is very natural. It means something like to have to leave / to have to get going.

You will often hear forms like:

• Ich muss los.
• Wir müssen bald los.
• Sie muss früh los.

It is especially common in spoken German and informal writing. It sounds much more idiomatic than trying to translate English word-for-word with something like gehen müssen in many everyday situations.

Here heute and früh are both adverbs:

• heute = today
• früh = early

Together they mean early today or, in natural English, often early today / early this morning, depending on context.

German often stacks time expressions without prepositions:

• heute Abend
• morgen früh
• gestern Nacht

So heute früh losmuss is a very normal way to say has to leave early today.

Because the sentence begins with a subordinate clause, and German uses commas to separate subordinate clauses from main clauses.

Structure:

• Da meine Schwester heute früh losmuss, = subordinate clause
• macht sie sich nur ein schnelles Rührei. = main clause

German punctuation is stricter here than English learners often expect: if you have a clause introduced by da, weil, dass, wenn, and so on, you normally need a comma.

Because German main clauses follow the verb-second rule.

The whole opening clause Da meine Schwester heute früh losmuss counts as the first element of the sentence. That means the finite verb of the main clause must come next:

• Da meine Schwester heute früh losmuss, macht sie ...

So the order is:

1. subordinate clause
2. finite verb: macht
3. subject: sie

If there were no opening clause, you would get the more familiar order:

• Sie macht sich nur ein schnelles Rührei.

Sich etwas machen means to make oneself something.

So:

• Sie macht sich ein schnelles Rührei = She makes herself a quick scrambled egg

Here:

• sich refers back to sie
• ein schnelles Rührei is the thing she makes

This is a very common German pattern for food and drinks:

• Ich mache mir einen Kaffee.
• Er macht sich ein Brot.
• Sie macht sich eine Suppe.

In this sentence, sich is functioning as a dative reflexive pronoun.

Because Rührei is a neuter noun:

• das Rührei

And here it is in the accusative singular as the direct object.

With an ein-word plus a neuter noun in nominative or accusative singular, the adjective takes -es:

• ein schnelles Rührei
• ein gutes Buch
• ein kaltes Getränk

So the form is exactly what you would expect from the adjective-ending pattern.

German usually treats Rührei as a singular food noun:

• das Rührei
• ein Rührei

So ein Rührei means one serving/dish of scrambled eggs.

This is just one of those places where German and English package the idea differently:

• English: scrambled eggs (plural)
• German: Rührei (usually singular)

It is completely normal German.

Here nur modifies ein schnelles Rührei.

So the idea is:

• because she has to leave early, she makes herself only a quick scrambled egg
• not a full breakfast, not several things, just that

Placing nur before the noun phrase makes that phrase the focus:

• macht sie sich nur ein schnelles Rührei

That is the natural position for nur in this sentence.