Sie näht den Saum mit der Nähmaschine um.

Because the verb is um|nähen, a separable verb.

In a normal main clause, the conjugated part goes in second position, and the prefix moves to the end:

• Sie näht den Saum um.

You will see the parts together again in forms like:

• Sie will den Saum umnähen.
• ..., weil sie den Saum umnäht.

So um is not randomly placed at the end; it belongs to the verb.

No. Here, um is not a preposition. It is the separable prefix of um|nähen.

That matters because:

• um as a preposition would normally be followed by a noun phrase, as in um den Tisch = around the table
• but in Sie näht den Saum mit der Nähmaschine um, the um stands alone because it belongs to the verb

So the structure is not näht ... around in a word-for-word English sense; it is one separable verb.

Because den Saum is the direct object, so it is in the accusative case.

The noun is:

• der Saum = nominative singular
• den Saum = accusative singular

Since Saum is masculine, the article changes from der to den in the accusative.

Compare:

• Der Saum ist schief. = The hem is crooked.
• Sie näht den Saum um. = She hems/sews the hem.

Because mit always takes the dative case.

The noun is:

• die Nähmaschine = nominative
• der Nähmaschine = dative

So after mit, you need:

• mit der Nähmaschine

This phrase expresses the instrument or means: she is doing it with the sewing machine.

näht is the 3rd person singular present tense form of nähen.

Conjugation of nähen in the present tense:

• ich nähe
• du nähst
• er/sie/es näht
• wir nähen
• ihr näht
• sie/Sie nähen

In this sentence, näht matches sie = she.

Here it means she.

At the beginning of a sentence, both sie and Sie are capitalized, so the capital letter alone does not tell you the meaning. The verb form does.

• Sie näht = she sews
• Sie nähen = you sew (formal)

Because the sentence has näht, the subject must be she, not formal you.

The dictionary form is um|nähen.

Dictionaries usually list separable verbs with the prefix attached, often marked like this:

• um|nähen

The sentence form näht ... um comes from that infinitive.

It can also help to recognize the simpler verb inside it:

• nähen = to sew
• um|nähen = to sew around / hem / resew, depending on context

Yes, German word order is somewhat flexible, but the main rule still holds:

• the conjugated verb stays in second position
• the separable prefix um stays at the end

So these are also possible:

• Mit der Nähmaschine näht sie den Saum um.
• Den Saum näht sie mit der Nähmaschine um.

These versions change the emphasis, but not the basic meaning.

It is a prepositional phrase that tells you how or with what she does the action.

So:

• den Saum = the thing being worked on; the direct object
• mit der Nähmaschine = the instrument

A good test is that you can remove it and the sentence still works:

• Sie näht den Saum um.

That shows mit der Nähmaschine is extra information, not the core object.

Because all nouns are capitalized in German.

So in this sentence:

• Saum is a noun
• Nähmaschine is a noun

That is why they begin with capital letters. This is a general rule in German, not something special about this sentence.

Yes. Nähmaschine is a compound noun:

• Näh- = sewing
• Maschine = machine

Together:

• Nähmaschine = sewing machine

German very often combines nouns into one longer word instead of using separate words the way English often does.