Moja siostra woli kanapkę z serem, ale ja wolę jajko i pomidor.

Because moja has to agree with siostra.

In Polish, possessive words like my change for:

• gender
• number
• case

Siostra is:

• feminine
• singular
• nominative here, because it is the subject

So the correct form is moja.

Compare:

• mój brat = my brother
• moja siostra = my sister
• moje dziecko = my child

Because they are in different cases.

• siostra is the subject of the sentence, so it stays in the nominative
• kanapkę is the direct object of woli, so it goes into the accusative

The basic noun is kanapka.
In the accusative singular, many feminine nouns ending in -a change to -ę:

• kanapka → kanapkę
• siostra → siostrę
• herbata → herbatę

So woli kanapkę means prefers a sandwich.

Because the verb changes depending on who is doing the action.

The verb is woleć = to prefer.

Here are the relevant forms:

• ja wolę = I prefer
• ty wolisz = you prefer
• on/ona/ono woli = he/she/it prefers

So:

• moja siostra woli = my sister prefers
• ja wolę = I prefer

That is true: Polish often omits subject pronouns because the verb form already shows the person.

So you could say:

• ale wolę jajko i pomidor

But ja is included here for contrast:

• My sister prefers..., but I prefer...

So ale ja wolę... sounds like:

• but I prefer...
• but as for me, I prefer...

It adds emphasis.

Z serem means with cheese.

The preposition z can mean different things depending on the case:

• z + instrumental = with
• z + genitive = from / off / out of

Here it means with, so ser becomes instrumental:

• ser → serem

So:

• kanapka z serem = sandwich with cheese
• kanapkę z serem = a sandwich with cheese / the sandwich with cheese, as the object

Because after z meaning with, Polish requires the instrumental case.

The noun ser in the instrumental singular is serem.

So:

• z serem = with cheese

You cannot say z ser.

A few more examples:

• z mlekiem = with milk
• z cukrem = with sugar
• z siostrą = with my sister

They are also direct objects, so they are in the accusative — but in these particular nouns, the accusative looks the same as the nominative.

That happens because:

• jajko is neuter
• pomidor is masculine inanimate

For many nouns in those groups, nominative = accusative in the singular.

So:

• jajko → jajko
• pomidor → pomidor

Compare that with a masculine animate noun:

• mam psa = I have a dog
  (pies → psa)

So the lack of change is normal here.

Only masculine animate nouns usually change in the singular accusative.

Polish masculine nouns are often divided into:

• masculine animate
• masculine inanimate

Pomidor is treated as masculine inanimate, so:

• nominative: pomidor
• accusative: pomidor

But with an animate noun:

• nominative: kot
• accusative: kota

So wolę pomidor is correct.

Not with exactly the same meaning.

• lubić = to like
• woleć = to prefer

This sentence is about preference, especially because the two halves are contrasted with ale.

So:

• Moja siostra lubi... = My sister likes...
• Moja siostra woli... = My sister prefers...

If you want the idea of choosing one thing rather than another, woleć is the better verb.

Because in Polish, ale normally introduces a contrasting clause, and it is preceded by a comma.

So:

• ..., ale ... = ..., but ...

This is standard punctuation in Polish.

Not completely. Polish word order is fairly flexible.

The given sentence has a very natural, neutral order:

• Moja siostra woli kanapkę z serem, ale ja wolę jajko i pomidor.

But other orders are possible for emphasis, for example:

• Kanapkę z serem woli moja siostra, ale ja wolę jajko i pomidor.

That version puts extra focus on kanapkę z serem.

For learners, the original word order is the safest and most natural to use.

Yes. Grammatically, it means egg and tomato as two coordinated nouns joined by i = and.

So the speaker is saying they prefer:

• jajko
• and
• pomidor

Depending on context, that could mean:

• egg and tomato as food items
• egg and tomato on a sandwich
• egg and tomato as part of a meal

The grammar itself simply shows a list of two things joined by i.