Learn Hardcore Turkish: Parkta stüdyo temalı bir sergi var. - There is a studio-themed exhibition in the park.

The suffix -ta is the locative case marker, which means “in” or “at.” In Turkish, the locative is normally -da, but because park ends with the voiceless consonant k, consonant harmony turns d into t, resulting in Parkta (“in the park”).

What role does bir play in bir sergi, and can it be omitted?

bir is the indefinite article meaning a or an. It marks sergi as a single, unspecified exhibition. In existential sentences like this one, including bir is very common. Technically you could say Parkta stüdyo temalı sergi var, but without bir the sentence sounds less specific and a bit awkward.

How is stüdyo temalı constructed, and what does it mean?

This is a compound adjective built from three parts:

stüdyo (studio)
tema (theme)
the adjectival suffix -lı (with)

So tema + -lı → temalı (“themed”), and stüdyo temalı literally means “with a studio theme,” i.e. studio-themed.

Why is var placed at the end, and what is its significance?

var is the existential verb in Turkish, meaning there is or there are. In an existential construction you normally follow this order:
location → noun phrase → var/yok.
Thus Parkta stüdyo temalı bir sergi var = “At the park a studio-themed exhibition exists.”

How would you ask “Is there a studio-themed exhibition in the park?” in Turkish?

Attach the question particle mı to var:
Parkta stüdyo temalı bir sergi var mı?

How do you express “There isn’t a studio-themed exhibition in the park”?

Use yok (the negative of var):
Parkta stüdyo temalı bir sergi yok.

How would you say “There are studio-themed exhibitions in the park”?

Make sergi plural (sergiler) and drop bir:
Parkta stüdyo temalı sergiler var.

Can you give a literal, word-for-word breakdown of the sentence?

Parkta – in/at the park
stüdyo – studio
temalı – themed (with a theme)
bir – a/an
sergi – exhibition
var – there is