Kupiłam nową obrożę dla psa, a stara leży teraz obok drzwi.

Kupiłam tells you that the speaker is female.

In the Polish past tense, the verb agrees with the gender of the speaker or subject:

• kupiłam = I bought (said by a woman)
• kupiłem = I bought (said by a man)

So this sentence is being said by a female speaker.

Also, kupiłam is first person singular past tense of kupić.

Because it is the direct object of the verb kupiłam.

The noun obroża is feminine, and in the singular its accusative form is:

• nominative: obroża
• accusative: obrożę

Since the sentence says I bought a new collar, the thing bought is the object, so Polish uses the accusative:

• Kupiłam obrożę = I bought a collar

Because the adjective has to agree with the noun in gender, number, and case.

Here:

• obroża is feminine singular
• it is in the accusative
• so the adjective nowy must also take the feminine singular accusative form

That gives:

• nowa obroża = a new collar (nominative)
• nową obrożę = a new collar (accusative)

Both words change because they work together as one noun phrase.

Because the preposition dla requires the genitive case.

The noun pies changes in the genitive singular to psa:

• nominative: pies = dog
• genitive: psa

So:

• dla psa = for the dog

This is very common in Polish: many prepositions force a specific case, and dla always takes the genitive.

Polish often omits a noun when it is already clear from context.

So stara here really means:

• stara obroża = the old collar

But since obroża was already mentioned in the first part of the sentence, repeating it is unnecessary. English does this too sometimes, as in the new one / the old one, but Polish can do it more directly with just the adjective.

So:

• a stara leży teraz obok drzwi
  means
• and the old one is now lying next to the door

Because a often links two clauses with a slight sense of contrast or switch of topic.

Compare:

• i = and in a simple additive sense
• a = and / while / whereas / as for

In this sentence:

• Kupiłam nową obrożę dla psa, a stara leży teraz obok drzwi.

The second part contrasts the new collar with the old one, so a sounds natural.

A rough feeling is:

• I bought a new collar for the dog, and the old one is now lying by the door.

Leży comes from leżeć, which means to lie or to be lying.

It is used for something that is in a resting position somewhere. So:

• stara leży teraz obok drzwi = the old one is lying now next to the door

Polish often prefers a more specific verb like leżeć rather than just a general to be.

Compare:

• jest obok drzwi = is next to the door
• leży obok drzwi = is lying next to the door

The second version gives a clearer physical picture.

The preposition obok takes the genitive case.

So the noun after it must be in the genitive:

• obok domu = next to the house
• obok stołu = next to the table
• obok drzwi = next to the door

In this sentence, drzwi happens to look the same in the nominative and genitive, which can be confusing. But grammatically, after obok, it is genitive.

Because drzwi is a plural-only noun in Polish.

This kind of noun is called pluralia tantum. Even though it refers to one object in English, in Polish it uses plural grammar.

So:

• drzwi = door
• te drzwi są otwarte = the door is open
  (literally, Polish treats it grammatically as plural: these door-PL are open-PL)

That is why learners often find drzwi surprising.

Kupiłam comes from kupić, which is perfective.

Perfective verbs in Polish usually show a completed action. So kupiłam means:

• I bought it / I have bought it
• the purchase is finished

The imperfective partner is kupować:

• kupowałam = I was buying / I used to buy / I was in the process of buying

In this sentence, the speaker is talking about a completed purchase, so kupiłam is the natural choice.

Yes. Polish word order is fairly flexible, because endings show grammatical roles.

The given sentence is natural:

• Kupiłam nową obrożę dla psa, a stara leży teraz obok drzwi.

But other orders are possible, for example:

• Teraz stara leży obok drzwi.
• Stara teraz leży obok drzwi.
• Obok drzwi leży teraz stara.

These versions shift the emphasis:

• teraz first emphasizes now
• obok drzwi first emphasizes the location
• stara first emphasizes the contrast with the new collar

So the original word order is not the only correct one, but it is a very normal neutral choice.

Yes. In Polish, when a connects two clauses, a comma is normally used.

So this is standard:

• Kupiłam nową obrożę dla psa, a stara leży teraz obok drzwi.

That comma helps separate the two parts:

1. Kupiłam nową obrożę dla psa
2. a stara leży teraz obok drzwi

This is one of the many cases where Polish punctuation is a bit stricter than English punctuation can be.