Learn Russian: Нам нужна кастрюля для супа.

In Russian, the "to need" construction is flipped compared to English. The person who needs something is put in the dative case (нам means "to us"), and the thing that is needed is the actual grammatical subject of the sentence. Literally, this sentence says "To us is needed a pot."

Why use the feminine form нужна here instead of the default нужно?

Нужен is a short adjective that must agree in gender and number with the thing being needed (the subject). Since кастрюля (pot) is a feminine noun, you must use the feminine form нужна. If you needed a masculine object like a knife (нож), it would be нам нужен нож.

Why does суп change to супа at the end of the sentence?

The preposition для (for / intended for) always requires the following noun to be in the genitive case. Because суп is a masculine noun ending in a consonant, you add -а to make it для супа.

Is the word order fixed, or could I say Кастрюля нужна нам для супа?

Russian word order is flexible, but the order in this sentence (нам нужна кастрюля) is the most natural and neutral way to state a need. Putting кастрюля at the very beginning would place special emphasis on it, sounding more like "A POT is what we need for the soup."