Dziewczynka z naszego sąsiedztwa lubi rysować koty i domy.

Polish does not have articles like English the and a/an.

So dziewczynka can mean the girl or a girl, depending on context. In a standalone example, English often has to choose one, but Polish leaves that unspecified unless the context makes it clear.

Not quite.

• dziewczynka = little girl or young girl
• dziewczyna = girl / young woman

So dziewczynka sounds younger and smaller. It is the natural word if you are talking about a child.

Here z means from, so it takes the genitive case.

That is why you get:

• dictionary form: sąsiedztwo
• after z meaning from: z sąsiedztwa

And because naszego modifies sąsiedztwa, it also has to be in the genitive:

• nasze sąsiedztwo
• z naszego sąsiedztwa

A very common point of confusion is that z can also mean with, and in that meaning it takes the instrumental. But here it means from, so it takes genitive, not instrumental.

Because naszego agrees with sąsiedztwa, which is in the genitive singular.

• nasze = basic form
• naszego = genitive singular form used here

You might expect naszym if you are thinking of phrases like z naszym kolegą (with our friend), but that is a different use of z. In this sentence, z means from, so it requires the genitive: naszego sąsiedztwa.

It would mean something slightly different.

• z naszego sąsiedztwa = from our neighborhood
• w naszym sąsiedztwie = in our neighborhood / in our vicinity

In this sentence, the phrase identifies which girl we mean: the one from our neighborhood. So z naszego sąsiedztwa is the natural choice.

Because Polish often uses a conjugated verb followed by an infinitive, just like English likes to draw.

Here:

• lubi = likes
  (3rd person singular present of lubić)
• rysować = to draw

So the pattern is:

• lubić + infinitive
• lubi rysować = likes to draw

This works the same way with many other verbs:

• chce rysować = wants to draw
• umie rysować = can draw / knows how to draw

Because lubić is the dictionary form, but in a sentence the verb has to be conjugated to match the subject.

The subject is dziewczynka, which is:

• 3rd person
• singular

So lubić becomes lubi.

A few forms of lubić are:

• lubię = I like
• lubisz = you like
• lubi = he/she/it likes

This is an aspect question.

• rysować = imperfective
• narysować = perfective

After lubić, Polish normally uses the imperfective when talking about something someone likes doing in general or habitually.

So:

• lubi rysować = she likes drawing / likes to draw

If you used narysować, it would sound more like liking the act of drawing something to completion, which is not the usual meaning here.

Yes. They are the direct objects of rysować, so they are in the accusative.

What can be confusing is that in Polish, the accusative sometimes looks exactly like the nominative.

Here:

• domy is accusative plural, and it looks the same as nominative plural
• koty is also accusative plural, and it also looks the same here

A useful thing to remember:

• in the plural, only masculine personal nouns have a special accusative that matches the genitive
• animals like koty are not masculine personal, so their plural accusative looks like the nominative

Compare:

• Widzę koty. = I see cats.
• Widzę studentów. = I see students.
  (studentów is masculine personal plural accusative)

Usually it sounds generic here: she likes drawing cats and houses as kinds of things.

That is very natural in Polish. Without extra context, it does not strongly suggest particular cats or particular houses.

If the speaker wanted to make them more specific, the sentence would usually need more context or additional words.

Yes, Polish word order is more flexible than English, because the endings show the grammatical relationships.

The version you have is a neutral, natural order:

• Dziewczynka z naszego sąsiedztwa lubi rysować koty i domy.

But other orders are possible for emphasis, for example:

• Koty i domy lubi rysować dziewczynka z naszego sąsiedztwa.
  This puts emphasis on cats and houses.

So the original sentence is best as a standard learning model, but it is not the only possible order.

Yes, several.

A few useful ones:

• cz in dziewczynka sounds like ch in church
• sz sounds like sh
• dz / dzi at the start of dziewczynka is a soft sound that English does not match exactly
• ą in sąsiedztwa is a nasal vowel, often sounding roughly like on/om
• w is pronounced like English v, except that at the end of a word or before a voiceless consonant it may sound more like f

Also, Polish stress is usually on the second-to-last syllable:

• dzieWCZYNka
• sąSIEDZtwa
• rySOwać

That regular stress pattern helps a lot when learning to say longer Polish words.