Moja kuzynka mieszka za granicą, ale często dzwoni do babci.

Because moja has to agree with kuzynka in gender, number, and case.

• kuzynka is feminine singular nominative
• so the possessive form must also be feminine singular nominative: moja

Compare:

• mój kuzyn = my male cousin
• moja kuzynka = my female cousin

English my never changes, but Polish possessives do.

Yes. Polish distinguishes the cousin’s gender:

• kuzyn = male cousin
• kuzynka = female cousin

That is different from English, where cousin can refer to either gender.

Polish often leaves out subject pronouns because the verb ending already shows the person and number.

Here:

• mieszka = he/she/it lives
• dzwoni = he/she/it calls / is calling

Since the sentence already has the subject moja kuzynka, adding ona would usually be unnecessary unless you want special emphasis.

Both are present tense, 3rd person singular forms:

• mieszkać → mieszka
• dzwonić → dzwoni

In this sentence, the present tense expresses a general fact or habit:

• she lives abroad
• she often calls grandma

So this is not just about what is happening right now, but about her usual situation and repeated action.

In za granicą, the noun granicą is in the instrumental case.

This whole expression is a very common Polish way to say abroad. Literally, it comes from the idea of being beyond the border, but learners should mostly treat za granicą as a fixed phrase meaning abroad.

So:

• mieszka za granicą = lives abroad

This is a very important distinction:

• za granicą = abroad, being abroad, location
• za granicę = abroad, going abroad, direction

Examples:

• Mieszka za granicą. = She lives abroad.
• Wyjechała za granicę. = She went abroad.

So the difference is similar to:

• location: where someone is
• motion: where someone is going

Because the verb dzwonić normally takes the preposition do when it means to call someone on the phone.

So Polish says:

• dzwonić do kogoś = to call someone

This is different from English, where call usually takes a direct object.

So:

• English: She calls grandma
• Polish: Ona dzwoni do babci

You can think of it as closer to she calls to grandma / she phones grandma, though the natural English translation is just calls grandma.

Because after do, Polish uses the genitive case.

The base form is:

• babcia = grandmother / grandma

After do, it becomes:

• do babci = to grandma / to grandmother

So the change is caused by case.

This is very common in Polish:

• do mamy = to mom
• do cioci = to aunt
• do kolegi = to a male friend

Często is an adverb meaning often. It tells you how frequently she calls.

So in:

• ale często dzwoni do babci

it modifies dzwoni.

Polish adverbs are usually easier than nouns and adjectives because they do not change for case, gender, or number.

Yes, Polish word order is more flexible than English, although some positions sound more neutral than others.

The version in your sentence:

• ale często dzwoni do babci

is very natural and neutral.

You may also hear:

• ale do babci często dzwoni
• ale dzwoni często do babci

These can sound more marked or emphasize a different part of the sentence. For learners, the original order is a good one to copy.

Because in Polish, ale normally takes a comma before it when it joins two clauses.

So:

• Moja kuzynka mieszka za granicą, ale często dzwoni do babci.

This is standard Polish punctuation.

It works similarly to English but, though Polish is often a bit stricter about using the comma here.

Yes. Dzwonić is imperfective, and that fits this sentence because the action is habitual or repeated.

• dzwoni = calls / is calling / calls regularly
• zadzwoni = will call once / will make a single completed call

So:

• często dzwoni do babci = she often calls grandma
• zadzwoni do babci = she will call grandma once

In habitual statements, Polish usually prefers the imperfective verb.

A few useful pronunciation points:

• cz sounds roughly like ch in chop
• ę and ą are nasal vowels, though their exact pronunciation changes a bit depending on the sounds around them
• ci before another vowel or at the end of a syllable gives a soft ć-type sound

Approximate guides:

• często ≈ CHEN-sto / CHEN-staw depending on accent and connected speech
• granicą ≈ gra-NEE-tson
• babci ≈ BAP-chee or BAB-chee with a very soft middle sound

The most important thing for a learner is to notice:

• cz is one sound
• ci is soft
• final ą is nasal

Yes, that is grammatically correct. It would still mean basically the same thing, but the emphasis changes.

• Moja kuzynka mieszka za granicą, ale często dzwoni do babci.
  This starts with the fact that she lives abroad, then adds the contrast that she still calls often.

• Moja kuzynka często dzwoni do babci, ale mieszka za granicą.
  This starts with the frequent calling, then adds the contrast that she lives abroad.

So both are possible; the original version highlights living abroad first.