Для котлет нужен хороший фарш, немного лука и терпение.

Для always takes the genitive case.

So котлет here is genitive plural of котлета.

• котлета = cutlet
• для котлеты = for a cutlet
• для котлет = for cutlets

In a cooking context, для котлет usually means for making cutlets or for cutlets as a dish.

That is a very common Russian pattern.

Many feminine nouns ending in -а form the genitive plural with a zero ending:

• котлета → котлет
• конфета → конфет
• машина → машин

So котлет is not the stem by itself in some random form; it is a normal genitive plural form.

That said, genitive plural is one of the trickier areas of Russian, so it is worth learning noun by noun and pattern by pattern.

Strictly speaking, нужен is the short-form adjective of нужный.

In sentences like this, it functions as the predicate and means is needed or is necessary.

It changes for gender and number:

• нужен = masculine singular
• нужна = feminine singular
• нужно = neuter singular
• нужны = plural

So in practice, it behaves a bit like English is needed.

Because нужен agrees with фарш, which is:

• masculine
• singular

So:

• фарш нужен
• соль нужна
• масло нужно
• яйца нужны

In this sentence, the predicate comes before a list, and it agrees with the first noun in that list: фарш.

Good question. Russian often allows the predicate before a list to agree with the nearest or first item, especially in practical styles like recipes or instructions.

So нужен хороший фарш, немного лука и терпение sounds natural because the first noun is фарш.

You may also hear or see нужны in similar sentences if the speaker wants to treat the whole list as a plural set. But the given sentence with нужен is perfectly idiomatic.

Because it is the thing that is needed.

With нужен / нужна / нужно / нужны, the thing needed is normally in the nominative:

• нужен фарш
• нужна соль
• нужно масло

If you add a person, that person is usually in the dative:

• Мне нужен фарш = I need mince
• Нам нужно терпение = We need patience

So in your sentence, фарш, терпение, and the phrase with немного are the needed items.

Because немного means a little, a bit of, or some, and it normally requires the genitive.

With a mass noun like лук, you get genitive singular:

• немного лука
• немного сахара
• немного сыра
• немного молока

So лука is genitive singular after немного.

After немного, лук would be grammatically wrong. You need the genitive: лука.

Also, in cooking Russian, лук often behaves like an ingredient noun, similar to English onion in add some onion. So немного лука means a little onion / some onion.

If you wanted to mean one whole onion, Russian would often say:

• одна луковица

So:

• немного лука = some onion
• одна луковица = one onion bulb

Терпение is nominative singular, neuter gender.

It is simply another item in the list of things that are needed:

• хороший фарш
• немного лука
• терпение

Russian can easily mix concrete things and abstract nouns in one list. So the sentence is a little playful or expressive: to make good cutlets, you need not only ingredients, but also patience.

No extra есть is needed.

Russian normally omits the present tense of to be in sentences like this, and нужен already works as the predicate.

So the sentence is complete as it stands:

• Для котлет нужен хороший фарш, немного лука и терпение.

Adding есть here would not be standard.

Yes. Russian word order is flexible.

Starting with Для котлет sets the context first: as for cutlets / for making cutlets. That is very natural in recipe-style language.

Other orders are possible, but they change the emphasis a little:

• Для котлет нужен хороший фарш, немного лука и терпение.
  Very natural recipe-style wording.

• Хороший фарш, немного лука и терпение нужны для котлет.
  More emphasis on the list itself.

So the original order is not the only possible one, but it is a very natural one.

Yes, but that is a slightly different style.

You may hear an impersonal recipe-style construction like:

• Для котлет нужно: хороший фарш, немного лука и терпение.

This feels more like To make cutlets, you need: followed by a list.

The version with нужен is more directly tied to the first noun фарш, while нужно makes the whole statement feel more impersonal and list-like. Both are possible in real Russian, but the sentence you were given is fully natural.