In unserem Abteil ist noch ein Platz am Fenster frei.

Abteil means a compartment or section in a train. In older-style trains, it can mean a closed compartment with several seats. In modern usage, it can also mean a seating section or area of the carriage.

So In unserem Abteil means in our compartment / in our section.

Because in here is showing location, not movement, so it takes the dative case.

• in unser Abteil would suggest movement into it in some contexts
• in unserem Abteil means in our compartment as a location

The noun Abteil is neuter, so the dative form of unser is unserem.

This is because German is a verb-second language in main clauses.

The first position in the sentence can be filled by many things, not just the subject. Here, In unserem Abteil is put first. Once that happens, the finite verb must come second:

• In unserem Abteil = first position
• ist = second position
• noch ein Platz am Fenster frei = the rest

So even though ein Platz is the subject, it comes after the verb.

Because ein Platz is the subject of the sentence, and subjects are in the nominative case.

The basic structure is:

• ein Platz = subject
• ist frei = is free / available

You would use einen Platz if it were a direct object, for example:

• Ich suche einen Platz. = I’m looking for a seat.

But here the sentence is saying that a seat exists/is available, so ein Platz is nominative.

Here noch means something like still or yet.

So the idea is:

• There is still a seat free
• There is one seat left available

It adds the sense that availability remains. Without noch, the sentence would simply state that a seat is free. With noch, it sounds more like there’s still one available.

Am is a contraction of an dem.

So:

• am Fenster = at the window / by the window

In natural English, this is often translated as window seat or seat by the window.

German often uses this kind of prepositional phrase where English might prefer a compound noun.

Because frei is being used predicatively, not directly before a noun.

Compare:

• ein freier Platz = a free seat
  Here frei comes before the noun and takes an ending.

• Der Platz ist frei. = The seat is free.
  Here frei comes after sein and does not take an ending.

In your sentence, frei describes ein Platz, but through the verb ist, so it goes in its plain form at the end.

Because it is not an attributive adjective here.

German adjectives have endings when they come before a noun:

• ein freier Platz

But after verbs like sein, werden, or bleiben, the adjective usually stays unchanged:

• Der Platz ist frei.

So in ist noch ein Platz am Fenster frei, frei has no ending because it is linked to the noun through ist.

Almost. In this sentence, Platz clearly means seat/place to sit.

German often uses Platz in travel situations where English would specifically say seat:

• Ist hier noch frei? = Is this seat free?
• Gibt es noch Plätze? = Are there any seats left?

Sitzplatz is more explicit and means seat or seating place, but Platz is very natural here.

Yes, that is possible and natural.

• In unserem Abteil ist noch ein Platz am Fenster frei.
• Es ist noch ein Platz am Fenster frei.

The version with In unserem Abteil puts the location first and sounds a bit more specific: In our compartment, there’s still a window seat free.

The version with Es ist ... is more like a general statement: There’s still a window seat free.

Yes, very often.

English commonly uses there is/there are. German often expresses the same idea without a special word like there and simply uses the verb with the noun:

• Es ist noch ein Platz frei.
• Noch ein Platz ist frei.
• In unserem Abteil ist noch ein Platz frei.

German can also use es gibt, but that is not always the best equivalent here. Es gibt means there is/there are in the sense of exists/available, while ist frei specifically means is free / unoccupied / available.

So ist ... frei is exactly right for an available seat.