Learn Hardcore Greek: Στη συνέντευξη για τη νέα δουλειά πρέπει να βάλεις υπογραφή σε συμβόλαιο. - At the interview for the new job you must put a signature on a contract.

So στη συνέντευξη literally means at the interview. In modern Greek this is normally written as one word (στη). Before some sounds and vowels you’ll also see στην (with -ν), e.g. στην αρχή.

Why is it στη συνέντευξη and not just σε συνέντευξη?

Greek normally uses the definite article for specific events.

στη συνέντευξη = at the interview (a particular one, which both speaker and listener know about)
σε συνέντευξη (without article) would sound more like at an interview / in an interview situation, more general or abstract, and is less natural here.

Since we are talking about the job interview you are going to, the article τη is expected.

How does για τη νέα δουλειά work? Is για necessary to express for the new job?

για means for / about and introduces the purpose or topic: η συνέντευξη για τη νέα δουλειά = the interview for the new job.
Without για, η συνέντευξη τη νέα δουλειά would be ungrammatical.

You could say something like η συνέντευξη της νέας δουλειάς (literally the new job’s interview), but for this meaning interview for the new job, για is the normal, natural preposition.

Why is the verb βάλεις and not βάζεις after πρέπει να?

After πρέπει να, Greek uses the subjunctive, not the present indicative.

βάζεις = you put (present indicative)
βάλεις = you put (aorist subjunctive)

With πρέπει να, the aorist subjunctive (πρέπει να βάλεις) is standard to express an obligation to do a single, complete action in the future: you have to sign once.

What does the expression να βάλεις υπογραφή literally mean, and is it the usual way to say to sign?

Literally, να βάλεις υπογραφή means to put (a) signature.
It is a very common idiomatic way to say to sign a document.

You will also hear the simple verb υπογράφω = to sign. Both are normal:

να βάλεις υπογραφή σε συμβόλαιο
να υπογράψεις συμβόλαιο

They mean essentially the same thing in this context.

What form is υπογραφή here (gender, number, case), and why is it that form?

υπογραφή is:

feminine
singular
accusative case

It is the direct object of the verb βάλεις (να βάλεις υπογραφή = to put a signature). The dictionary form η υπογραφή (nominative) looks the same as the accusative in the singular; the role in the sentence tells you it’s the object.

Why is there no article before συμβόλαιο? Could I say σε ένα συμβόλαιο or στο συμβόλαιο instead?

Without an article, σε συμβόλαιο here means on a contract / in a contract in an indefinite way: not a specific contract already known, just some contract.

You can say:

σε ένα συμβόλαιο = on a contract (still indefinite, but more explicitly a contract)
στο συμβόλαιο = on the contract (a particular, known contract)

All are grammatically correct; the original simply keeps it indefinite and neutral.

Could we use υπογράψεις instead of βάλεις υπογραφή? Does it change the meaning?

Yes, you can say:
Στη συνέντευξη για τη νέα δουλειά πρέπει να υπογράψεις συμβόλαιο.

This is fully natural and means the same thing: At the interview for the new job you have to sign a contract.
βάλεις υπογραφή and υπογράψεις are stylistic variants here; βάλεις υπογραφή is a bit more idiomatic/colloquial, υπογράψεις is slightly more compact.

Can the word order change, for example to Πρέπει να βάλεις υπογραφή σε συμβόλαιο στη συνέντευξη για τη νέα δουλειά?

Yes. Greek word order is quite flexible.

Πρέπει να βάλεις υπογραφή σε συμβόλαιο στη συνέντευξη για τη νέα δουλειά is grammatically correct and natural.
The original order Στη συνέντευξη... πρέπει να βάλεις... puts a little more emphasis on at the interview by mentioning it first, but the basic meaning is the same.

How would I say this more politely or formally, for example to someone I address with formal you?

In formal speech you use the second person plural for you:
Στη συνέντευξη για τη νέα δουλειά θα πρέπει να βάλετε υπογραφή σε συμβόλαιο.

Changes:

βάλεις → βάλετε (2nd person plural)
Often θα πρέπει is used instead of plain πρέπει, which can sound slightly more polite/soft.

How do you pronounce συνέντευξη, δουλειά, and συμβόλαιο?

Approximate pronunciation (stressed syllable in capitals):

συνέντευξη → si-NEN-tev-ksee (more precisely [siˈnenðevksi] or [siˈnenðuksi])
δουλειά → dhou-LIA (with soft λι; roughly [ðuˈʎa] or [ðuˈlja])
συμβόλαιο → sim-VO-le-o (often [simˈvɔleo] or [simˈvole.o])

Note: δ is like English th in this; ξ is like ks in box.

Does δουλειά here mean job or work, and what’s the difference from εργασία?

In this sentence, δουλειά clearly means job / position: τη νέα δουλειά = the new job you are getting hired for.

δουλειά is the everyday, informal word and can mean:

a job / post (Βρήκα δουλειά. = I found a job.)
work in general (Έχω πολλή δουλειά. = I have a lot of work.)

εργασία is more formal and is used in official, academic, or bureaucratic contexts (contracts, laws, studies, etc.). Here, δουλειά is the natural choice in normal conversation.