Learn Hardcore Turkish: Film başlamadan önce biletler tükenmişti. - The tickets had sold out before the film started.

You say film başlamadan önce. This breaks down as follows:
• film = “the film” (nominative)
• başlamak = “to start” → root başla-
• -ma- = negation of the verb (“not doing X”)
• -dan = ablative case (“from”)
Together başla-ma-dan literally “from not starting,” and adding önce (“before”) gives “before the film started.”

What exactly does the suffix -madan indicate?

-madan is the negative gerund (adverbial) form of a verb. It combines:

-ma/-me (negation)
-dan/-den (ablative case)
When you attach it to a verb root and follow with önce, X-madan önce means “before doing X.”

Why is tükenmişti used here, and what tense/aspect is it?

tükenmişti is formed with the evidential/past participle suffix -miş plus the simple past -ti. It expresses that the action (tickets running out) was completed prior to another past event (the film’s start). In English it corresponds to the past perfect “had sold out.”

What’s the difference between tükenmek, satılmak, and bitmek when talking about tickets?

• tükenmek = “to run out” or “to be sold out.” It’s the standard verb for stock running out (tickets, supplies, etc.).
• satılmak = “to be sold” (passive of “to sell”), but doesn’t imply exhaustion of stock.
• bitmek = “to finish/end,” and sometimes “to run out,” but tükenmek is more precise for “sold out.”
So for tickets you normally use tükenmek.

Why is there no pronoun for “they” in tükenmişti?

Turkish verbs mark person and number, and when the subject is clear (here biletler “tickets”), you omit the pronoun. The verb ending -ti (plus evidential -miş) already implies third-person plural.

Why does the verb appear at the end of the clause?

Turkish is an SOV (Subject-Object-Verb) language. In both simple and complex sentences the finite verb typically comes last.

Can we change the word order of this sentence?

Yes. Turkish word order is relatively flexible for emphasis. You could say:
Biletler film başlamadan önce tükenmişti.
All versions are grammatically correct, but placing the time clause first (“film başlamadan önce”) is very natural.

Could we say filmden önce biletler tükenmişti instead of başlamadan önce?

filmden önce biletler tükenmişti literally “before the film, the tickets had sold out.” It’s acceptable if you mean “before the film screening” in general (e.g. before the showtime). But if you specifically want “before it started,” use başlamadan önce to refer to the moment the film begins.