Learn Hardcore Turkish: Biberiyeli fırın yemeklerinin kokusu tüm mahalleyi sarıyordu. - The aroma of rosemary-baked dishes was enveloping the entire neighborhood.

biberiye → biberiyeli (“with rosemary” or “rosemary-flavored”)
Because biberiye ends in a vowel, a buffer consonant -y- is inserted before -li: biberiye + y + li → biberiyeli.
The suffix -li/-lı/-lu/-lü attaches to nouns to mean “with X” or “having X,” following Turkish vowel harmony.

What are fırın yemekleri? Could I say fırında yemekler instead?

fırın = “oven”
yemek = “food” or “dish”
yemekler = plural “dishes”

So fırın yemekleri literally means “oven dishes” or “what you cook in the oven” (e.g. casseroles, baked goods).
If you say fırında yemekler, it’s grammatically possible (“dishes in the oven”) but less idiomatic as a compound. fırın yemekleri is the set phrase for “oven-baked dishes.”

Why is yemeklerinin in the genitive case, and what role does kokusu play?

The phrase yemeklerinin kokusu means “the smell of the dishes.” In Turkish possession structures (X’in Y’su):

The possessor (X = fırın yemekleri) takes the genitive suffix -nin → yemeklerin.
The possessed noun (Y = koku, “smell”) takes the third-person possessive suffix -su (vowel-harmonized) → kokusu.

So:
fırın yemekleri + -nin → fırın yemeklerinin (of the oven dishes)
koku + -su → kokusu (their smell)

Why does kokusu have a possessive suffix when it’s the sentence’s subject?

Here kokusu is “the smell of the dishes,” so it’s the possessed noun in the genitive-possessive construction. Even as the grammatical subject of sarıyordu, it must show possession:

koku (smell) + -su (3rd-person singular possessive) = kokusu.
Without -su it would mean “a smell,” not “the dishes’ smell.”

Why is the verb sarıyordu in the past continuous rather than the simple past?

sarmak means “to wrap” or “to envelop.”

Present continuous: sarıyor “is enveloping”
Past continuous: sarıyordu “was enveloping”

The suffixes break down as: root sar- + -ıyor (continuous) + -du (past).
Using past continuous emphasizes an ongoing action in the past (“the smell kept spreading all over...”); the simple past sardı would sound more like a single completed event.

Why does tüm mahalleyi take the accusative suffix -yi?

The verb sarmak here is transitive: “to envelop/cover something.” When the object is definite or whole (“the entire neighborhood”), Turkish marks it with the accusative suffix.

mahalle = “neighborhood”
tüm = “entire/whole”
mahalle
- -yi = mahalleyi (“the neighborhood” as a definite object)

What is the role of tüm and can I use bütün instead?

Both tüm and bütün mean “whole” or “entire.”

tüm mahalleyi = “the whole neighborhood”
You could say bütün mahalleyi with no significant change in meaning.
Choice between them is often a matter of style or emphasis.

If there were only one dish instead of many, how would the sentence change?

You’d switch yemekleri (plural) to yemeğini (singular genitive):
Biberiyeli fırın yemeğinin kokusu tüm mahalleyi sarıyordu.
Here:

yemeğin = “of the dish” (singular)
-in = genitive
Everything else (possessive on koku, accusative on mahalle) stays the same.