Learn Hardcore Turkish: Uzak köyde sadece bir market var. - There is only one market in the distant village.

In Turkish var is the existential verb meaning “there is” or “there are.” You use it to state that something exists or is present. It never takes a subject in the nominative with agreement; instead, the thing that exists simply appears before var in the nominative case. Its negative counterpart is yok (“there isn’t / there aren’t”).

What case is köyde, and how is it formed?

Köyde is in the locative case, which means “in/at the village.” You form it by attaching the locative suffix -de or -da (depending on vowel harmony) directly to the noun. Since köy has the front vowel ö, we use -de, giving köy + de → köyde. No apostrophe is used because köy is a common noun.

Why doesn't uzak get the locative suffix in uzak köyde?

Case suffixes always attach to the noun itself, not to any preceding adjectives. Here, köy is the noun that takes the locative -de, so the adjective uzak stays unchanged and the suffix goes on köy.

Why is there no bir before uzak köyde, but there is one before market?

The indefinite article bir (“a/one”) is optional for location phrases and often dropped when the emphasis is purely on the place.

Uzak köyde can mean “in the remote village” (definite) or simply “in a remote village” (indefinite), depending on context.
If you want to emphasize “in a remote village,” you could say uzak bir köyde.
For market, we specify “one store,” so bir market clearly means “a (single) market.”

What does sadece mean, and is it necessary here?

Sadece means “only” or “just.” It restricts the meaning, so sadece bir market var tells us “there is only one market and nothing else.” Without it, Uzak köyde bir market var still means “In the remote village there is a market,” but you lose the nuance that it’s the only one.

Why is the verb var placed at the end of the sentence? Is that always the case?

Turkish generally follows Subject–Object–Verb (SOV) order, and existential sentences place the location and any modifiers first, then the noun phrase, and finally var or yok. In neutral style you’ll almost always find var at the end:
“[Location] + [Modifier] + [Noun Phrase] + var.”

Could I use yalnızca instead of sadece?

Yes. Yalnızca is a synonym of sadece meaning “only.” It’s slightly more formal or literary, but you can freely replace one with the other:
“Uzak köyde yalnızca bir market var.”

Where should sadece be placed if I want to emphasize different parts of the sentence?

To focus on the fact that there’s only one market, place it before bir market:
“Uzak köyde sadece bir market var.”
To stress that only in the remote village exists a market (implying other places have none), put it at the very beginning:
“Sadece uzak köyde bir market var.”
Moving sadece changes the emphasis, but it always stays before the element it modifies.