На куртке не хватает одной пуговицы, и я ищу новую.

Because the button is understood as something attached on the jacket, not something located inside it.

• на куртке = on the jacket
• в куртке = in the jacket

Russian often uses на for things that are on a surface or fastened to clothing: buttons, pockets, badges, stains, and so on.

Не хватает is from не хватать, which often means to be lacking, to be missing, or not to be enough.

So here it literally means something like:

• On the jacket, one button is lacking

In more natural English, we say One button is missing from the jacket.

Because не хватает suggests that something which should be there is absent.

• нет = there is no / there isn’t
• не хватает = something is missing; there is a shortage/lack

For a jacket, buttons are expected parts of it, so не хватает is very natural. It implies an incomplete set.

Because after не хватает, the thing that is lacking normally goes in the genitive case.

So:

• пуговица → nominative
• пуговицы → genitive singular

And одной matches пуговицы in case, gender, and number:

• feminine
• singular
• genitive

So одной пуговицы = of one button grammatically, even though English translates it simply as one button.

Because одну would be accusative, but не хватает requires the genitive for the missing thing.

Compare:

• Я вижу одну пуговицу = I see one button.
  → direct object, so accusative
• Не хватает одной пуговицы = One button is missing.
  → after не хватает, so genitive

Yes, you can.

That would still mean A button is missing from the jacket.

Adding одной makes the number explicit: exactly one button is missing. Without it, the idea is a bit less specific, though in many situations the meaning is still clear.

Because Russian often omits a noun when it is obvious from context.

Here новую really means:

• новую пуговицу = a new button

Since пуговицу is already understood, the adjective can stand on its own, like English a new one.

Because the hidden noun is пуговицу, which is:

• feminine
• singular
• accusative

Ищу takes a direct object, so Russian uses the accusative:

• пуговица → пуговицу
• новая → новую

So ищу новую means I’m looking for a new one, with пуговицу understood.

Because ищу is the imperfective present tense and describes the process of searching.

• ищу = I am looking for / I look for
• поищу = I’ll look for
• найду = I’ll find

In this sentence, the speaker is describing the current situation: a button is missing, and they are searching for a replacement. So ищу is the natural choice.

Russian и can simply connect two related facts, even when English might prefer and or so.

So:

• ..., и я ищу новую = ..., and I’m looking for a new one
• depending on context, English might naturally say ..., so I’m looking for a new one

If you want to make the cause-and-effect relationship more explicit in Russian, you could say:

• На куртке не хватает одной пуговицы, поэтому я ищу новую.

Because this is a compound sentence with two clauses:

1. На куртке не хватает одной пуговицы
2. я ищу новую

Each clause has its own predicate:

• не хватает
• ищу

So Russian separates them with a comma before и.

Yes, that is also possible, but the nuance is a little different.

• на куртке focuses on the physical location: the button is on the jacket
• у куртки treats the jacket more like the thing that possesses the button

Both can be understood, but на куртке is especially natural when talking about something attached to the outside of the clothing.

No, Russian word order is fairly flexible because the case endings show the grammatical roles.

This order is natural because it presents the situation first and then the reaction:

• На куртке не хватает одной пуговицы = setting the scene
• и я ищу новую = what the speaker is doing about it

You could rearrange parts for emphasis, but the given order sounds smooth and idiomatic.