Перед встречей она долго смотрела на свои брови и сказала, что ресницы сегодня выглядят лучше, чем обычно.

Because the preposition перед normally takes the instrumental case when it means before or in front of.

So:

• встреча = meeting
• instrumental singular: встречей

That gives перед встречей = before the meeting.

This is true for time expressions too, not only physical location.

In Russian, past-tense verbs agree with the subject in gender and number.

The subject here is она, so the verbs must be feminine singular:

• он смотрел / сказал
• она смотрела / сказала
• оно смотрело / сказало
• они смотрели / сказали

So смотрела and сказала both match она.

Because the verb смотреть in the sense of to look at is normally used with на + accusative.

The pattern is:

• смотреть на кого?
• смотреть на что?

So:

• смотрела на свои брови = looked at her eyebrows

Without на, the sentence would be ungrammatical in standard Russian with this meaning.

Russian usually uses the reflexive possessive свой when the possessor is the same as the subject.

Here, the subject is она, and the eyebrows belong to that same person, so свои is the natural choice:

• она смотрела на свои брови = she looked at her own eyebrows

If you used её, it could sound less natural here, and it may also suggest someone else’s perspective or create ambiguity.

A good rule:

• if the subject owns the thing, Russian often prefers свой
• if someone else owns it, use его / её / их

Because свои has to agree with брови.

Here is what matters:

• брови is plural
• it is the object of смотрела на, so it is in the accusative
• for inanimate plural nouns, accusative plural looks the same as nominative plural

So we get:

• свои брови

This is the plural form of свой agreeing with a plural noun.

Because Russian, like English, often refers to paired body parts in the plural when talking about them generally.

So:

• брови = eyebrows
• ресницы = eyelashes

You can use the singular if you mean just one:

• бровь = eyebrow
• ресница = eyelash

But in this sentence, the speaker is talking about them as a pair or as a set, so plural is the normal choice.

Because что introduces a subordinate clause after сказала.

The structure is:

• она сказала, что...
• she said that...

In Russian, a clause introduced by что is normally separated by a comma.

So:

• она сказала, что ресницы сегодня выглядят лучше...

That comma is required.

Because the subject of that clause is ресницы, which is plural.

So the verb must also be plural:

• ресница выглядит = the eyelash looks
• ресницы выглядят = the eyelashes look

This is the standard 3rd person plural present-tense form of выглядеть.

This is a comparative structure.

• лучше = better
• чем = than
• обычно = usually / as usual

So лучше, чем обычно means better than usual or more literally better than they usually do.

Russian often leaves out repeated words that are understood from context. The fuller idea would be something like:

• лучше, чем обычно выглядят

But Russian normally omits that repeated verb.

Here чем обычно functions as a comparative clause, and Russian normally puts a comma before чем in this kind of structure.

So:

• лучше, чем обычно

This is very common after comparatives such as:

• больше, чем...
• меньше, чем...
• лучше, чем...
• хуже, чем...

Долго means for a long time.

So:

• она долго смотрела = she looked for a long time

Its position is normal and natural. Russian adverbs are fairly flexible, but placing долго before the verb is a very common way to say it.

You could also see:

• она смотрела долго

That is also possible, but the original version sounds very natural.

Because сегодня modifies выглядят and tells us when they look better.

So the meaning is:

• her eyelashes today look better than usual

Its placement is natural because it stays close to the part of the sentence it belongs to:

• что ресницы сегодня выглядят лучше...

Russian word order is flexible, but this version is clear and idiomatic.

Yes. Russian word order is more flexible than English word order, because case endings already show many grammatical relationships.

For example, you could move some parts around for emphasis. But the original order is good because:

• Перед встречей sets the time right away
• она then gives the subject
• долго смотрела на свои брови describes the action
• и сказала, что... continues naturally into reported speech

So even though other orders are possible, this one sounds smooth and neutral.

Yes, that is possible, but it is slightly different in structure.

• лучше, чем обычно = better than usual
• лучше обычного = literally better than the usual/norm

Both can work in many contexts. The version in your sentence uses the very common comparative pattern лучше, чем обычно, which is especially transparent for learners because it closely matches English better than usual.