Моя попутчица сидела у иллюминатора и всё время смотрела на облака.

Because попутчица is a feminine noun meaning female traveling companion / fellow passenger. The possessive pronoun has to agree with it, so you get:

• мой попутчик = my male fellow traveler
• моя попутчица = my female fellow traveler

So моя is feminine singular nominative, matching попутчица.

Попутчица means a woman traveling with you or in the same direction as you: a female fellow traveler, female traveling companion, or sometimes female seatmate / fellow passenger, depending on context.

It comes from the idea of someone who is on the same route/path as you.

Its masculine equivalent is попутчик.

Those are past tense feminine singular forms.

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

• сидел / смотрел = he was sitting / he was looking
• сидела / смотрела = she was sitting / she was looking
• сидело / смотрело = it was sitting / it was looking
• сидели / смотрели = they were sitting / they were looking

Since the subject is попутчица (feminine), the verbs are feminine too.

Иллюминатора is in the genitive singular.

The preposition у commonly means by, near, next to and requires the genitive case:

• у окна = by the window
• у двери = by the door
• у иллюминатора = by the porthole / airplane window

So сидела у иллюминатора means was sitting by the window.

Literally, иллюминатор is a porthole or a round window, especially on a ship or airplane.

In this sentence, it refers to the airplane window. So сидела у иллюминатора means she was sitting in the seat by the window.

Because the verb смотреть на takes the accusative case for the thing you are looking at.

So:

• смотреть на небо = to look at the sky
• смотреть на море = to look at the sea
• смотреть на облака = to look at the clouds

Here облака is the accusative plural form of облако.

Even though на can sometimes suggest movement onto something, with смотреть на it simply means look at and still requires accusative.

Here it is accusative plural, because it is the object of смотрела на.

But for this noun, the nominative plural and accusative plural are the same:

• nominative plural: облака
• accusative plural: облака

So the form looks identical, but the function in the sentence is accusative.

Всё время means all the time, the whole time, or continuously in this context.

So:

• всё время смотрела на облака = she kept looking at the clouds the whole time

A useful thing to notice is the spelling всё with ё. In many printed texts, ё is often written as е, so you may also see все время, but here the meaning is still all the time, not everyone.

Because the sentence describes an ongoing scene in the past, not completed single actions.

• сидела = was sitting
• смотрела = was looking / kept looking

Imperfective is used here because the speaker is describing what was happening over a period of time.

If you used perfective verbs, the meaning would shift toward completed actions, which would not fit this calm descriptive scene as well.

Yes. Russian word order is relatively flexible, though the neutral version here is very natural:

• Моя попутчица сидела у иллюминатора и всё время смотрела на облака.

You could rearrange parts for emphasis, for example:

• У иллюминатора сидела моя попутчица...
  This emphasizes where she was sitting.
• На облака моя попутчица смотрела всё время.
  This sounds more marked and emphasizes the clouds.

So the original order is the most straightforward and neutral.

Yes. It is fairly movable, though some positions sound more natural than others.

For example:

• ...и всё время смотрела на облака.
• ...и смотрела всё время на облака.
• ...и смотрела на облака всё время.

All are understandable, but the first version is the most natural and standard here.

Placing всё время before the verb often makes the sense of continuously / the whole time especially clear.

Yes. И simply means and, linking two actions done by the same person:

• she was sitting by the window
• and was looking at the clouds the whole time

Because both verbs refer to the same subject, Russian does not repeat the subject:

• Моя попутчица сидела... и смотрела...

That is exactly like English: My fellow traveler was sitting... and looking... / ...and she kept looking...