Learn Hardcore Turkish: Şehir dışında arazi var.

In Turkish, -da/-de is the locative case ending meaning in, on, or at. Because dış ends with a consonant, a buffer n is inserted before the suffix, and vowel harmony turns it into -ında. So dış+ında becomes dışında, literally at the outside. When you put şehir in front, şehir dışında means outside the city (on the outskirts).

Why is şehir not in the genitive form (şehrin) here, and how does şehrin dışında differ from şehir dışında?

Şehrin dışında uses the genitive–locative construction (şehir + -in + -de) to mean outside of the city in a literal sense. By contrast, şehir dışı is a fixed compound meaning outskirts, and its locative is şehir dışında. Both forms are correct:

şehrin dışında = “outside of the city” (more literal)
şehir dışında = “in the outskirts” (set expression)

What exactly is var, and why does it appear at the end of the sentence?

Var is an impersonal existential verb that means there is/there are. Turkish word order places the verb at the end, so Şehir dışında arazi var translates word-for-word as Outside the city land there is.

Why doesn’t Turkish use articles like a or the before arazi?

Turkish has no direct equivalents for indefinite or definite articles like a and the. Nouns stand alone, and context determines whether they are definite or indefinite. So arazi can mean “land,” “a piece of land,” or “the land,” depending on the situation.

How do you say “there are lands outside the city” (i.e. in the plural)?

Attach the plural suffix -ler/-lar to arazi before var:
Şehir dışında araziler var
This means there are lands outside the city. Note that arazi often functions as a mass noun, so the singular form is more natural unless you specifically refer to multiple separate plots.

How would you turn Şehir dışında arazi var into a yes/no question?

Add the question particle -mı/-mi/-mu/-mü after var (matching vowel harmony):
Şehir dışında arazi var mı?
This means Is there land outside the city?

Is it correct to say Arazi şehir dışında var, or is the word order fixed?

Turkish word order is flexible, but the most natural order puts the locative phrase before the noun:
Şehir dışında arazi var.
Arazi şehir dışında var is understandable, but less common.

Can arazi be considered a countable noun in Turkish?

Arazi is usually an uncountable (mass) noun meaning land in general. If you want to talk about distinct parcels or lots, you can pluralize it (araziler), but in everyday use arazi remains singular for a general sense of land.