Στο ταμείο χρησιμοποίησα ένα κουπόνι και πήρα έκπτωση.

Στο is a contraction of σε + το (at/in/to + the). In this sentence it means at the (as in at the checkout/cashier).
After σε, the noun goes in the accusative case, so ταμείο is accusative singular: (το) ταμείο → στο ταμείο.

Ταμείο can mean several related things: cash register, checkout, cash desk, sometimes fund/treasury, and in some contexts even bank/till.
Here, Στο ταμείο is a very common set phrase meaning at the checkout / at the cashier (in a store). Context (using a coupon and getting a discount) strongly points to that meaning.

Χρησιμοποίησα is simple past (aorist), 1st person singular: I used.
Greek often uses the aorist for a completed, one-time action in the past—exactly what “I used a coupon” describes.

The dictionary form is χρησιμοποιώ (I use).
So: χρησιμοποιώ (present) → χρησιμοποίησα (aorist/simple past).

Ένα means a/one, so ένα κουπόνι = a coupon (introducing it as new information).
Το κουπόνι would mean the coupon, implying a specific coupon already known from context.

Yes, και means and. It links two actions: 1) χρησιμοποίησα ένα κουπόνι = I used a coupon
2) πήρα έκπτωση = I got a discount
Greek commonly strings actions together with και in a natural, straightforward way.

Πήρα is the aorist of παίρνω. Literally it’s I took / I received, and very often it corresponds to English I got depending on context.
With έκπτωση (discount), Greek uses παίρνω έκπτωση = to get/receive a discount.

In Greek, it’s common to omit the article when talking about an amount/type of something in a general way, especially with fixed expressions.
πήρα έκπτωση is a very natural phrase meaning I got a discount (not focusing on “one discount” as a countable item).
You can say πήρα μια έκπτωση, but it can sound more specific/emphatic (as if highlighting a particular discount).

Έκπτωση is accusative singular because it’s the direct object of πήρα (I got what? → έκπτωση).
The nominative is η έκπτωση; the accusative is την έκπτωση (often without the article, as here).

Yes, word order is fairly flexible.
Στο ταμείο χρησιμοποίησα ένα κουπόνι... puts emphasis on where/when (at the checkout).
Χρησιμοποίησα ένα κουπόνι στο ταμείο... is also correct and a bit more neutral, focusing first on the action.

Στο is pronounced like sto (one syllable).
ταμείο is ta-MI-o (three syllables).
Together: sto ta-MI-o.
No special sound change is required beyond normal connected speech.