Mit der Bürste bekomme ich den Fleck aus der Jacke schneller weg.

Because mit always takes the dative case.

• die Bürste = nominative
• der Bürste = dative

So after mit, you must say mit der Bürste.

This phrase means with the brush, and it tells you the tool/instrument used.

This is the separable verb wegbekommen.

In a main clause, separable verbs split:

• Ich bekomme den Fleck weg.
• infinitive: den Fleck wegbekommen

So:

• bekomme = the conjugated part, in second position
• weg = the separable prefix, moved to the end

Here wegbekommen means something like manage to get rid of / remove. It is very common in everyday German.

Not by itself, no. In this sentence you should understand wegbekommen as a whole.

• bekommen alone often means to get / receive
• wegbekommen means to get something away / remove it / get rid of it

So this is not about receiving a stain or a brush. It is an idiomatic separable verb.

Because den Fleck is the direct object of wegbekommen.

The basic idea is:

• ich bekomme etwas weg
• what do I remove? → den Fleck

Since Fleck is masculine (der Fleck), its accusative form is den Fleck.

• nominative: der Fleck
• accusative: den Fleck

Because aus means out of / from inside and it takes the dative case.

Here the stain is being understood as something that is in the fabric / in the jacket, so German naturally says aus der Jacke.

• aus der Jacke = out of the jacket

This is a very common way to talk about removing stains.

Both can sometimes make sense, but they are not exactly the same.

• aus der Jacke suggests getting the stain out of the material/fabric
• von der Jacke suggests getting something off the surface

With stains, German often prefers aus because the stain is thought of as being in the cloth, not just sitting on top of it.

So den Fleck aus der Jacke bekommen sounds very natural.

Because aus requires the dative case.

• nominative: die Jacke
• dative: der Jacke

So:

• aus die Jacke = incorrect
• aus der Jacke = correct

Schneller is the comparative of schnell:

• schnell = fast / quickly
• schneller = faster / more quickly

There is no als because the comparison is only implied, not stated explicitly.

So schneller here means something like:

• more quickly
• faster
• faster than with some other method / than before / than otherwise

German often uses comparatives this way without saying exactly what the comparison is.

German main clauses follow the verb-second rule: the conjugated verb must be in second position.

If you put Mit der Bürste first, then bekomme must come immediately after it:

• Mit der Bürste bekomme ich den Fleck ... weg.

This is called topicalization. The speaker puts Mit der Bürste first to highlight the method or tool.

You could also say:

• Ich bekomme den Fleck mit der Bürste schneller aus der Jacke weg.

That is also possible, but starting with Mit der Bürste gives that part more emphasis.

Because in a main clause, the prefix of a separable verb usually goes to the end of the clause.

Everything else comes before it, including objects, prepositional phrases, and adverbs:

• Mit der Bürste bekomme ich den Fleck aus der Jacke schneller weg.

So weg acts almost like a bracket closing the whole verbal idea.

This is very normal with separable verbs:

• Ich mache die Tür auf.
• Er räumt das Zimmer auf.
• Wir bekommen den Fleck weg.

Yes, it sounds quite natural and everyday.

Especially den Fleck wegbekommen is a common spoken-style way to say remove the stain.

A more formal or neutral alternative would be something like:

• Mit der Bürste kann ich den Fleck schneller aus der Jacke entfernen.

But the original sentence sounds very natural in conversation.

Yes. German word order is flexible, as long as the grammar rules are respected.

For example, you could also hear:

• Ich bekomme den Fleck mit der Bürste schneller aus der Jacke weg.
• Den Fleck bekomme ich mit der Bürste schneller aus der Jacke weg.

These versions shift the emphasis:

• Mit der Bürste ... emphasizes the tool
• Den Fleck ... emphasizes what is being removed
• Ich ... is the more neutral starting point

But in all of them, the finite verb stays in second position and weg stays at the end.

A useful breakdown is:

• Mit der Bürste = prepositional phrase showing the tool
• bekomme = finite verb
• ich = subject
• den Fleck = direct object
• aus der Jacke = where it is being removed from
• schneller = adverb/comparative
• weg = separable prefix

So the sentence is built around the idea:

• I get/remove the stain out of the jacket faster, with the brush

That helps show why the cases and word order look the way they do.