Pies podał mi łapę, a potem położył głowę na moim kolanie.

Polish does not have articles, so it does not use separate words like a or the.

So:

• pies can mean a dog or the dog
• łapę can mean a paw or the paw
• głowę can mean a head or the head

The context tells you which meaning is intended.

Literally, it means the dog gave me a paw or gave me its paw.

In natural English, this usually means the dog offered its paw or gave me its paw to shake. It is a very normal expression in Polish when talking about a dog doing that trick or affectionate gesture.

• podać = to hand, give, offer
• łapa = paw
• podać łapę = to give a paw

Both mean to me, but mi is the normal unstressed form here.

• mi = common, short, unemphatic form
• mnie = fuller form, used for emphasis, contrast, or after prepositions

So:

• Pies podał mi łapę = neutral, natural
• Pies podał mnie łapę = incorrect
• Pies podał łapę mnie = possible only with special emphasis, like to me, not someone else

Grammatically, mi is in the dative case, because it is the indirect object: the dog gave the paw to me.

Because they are in the accusative singular, not the nominative.

Their dictionary forms are:

• łapa = paw
• głowa = head

But in the sentence they are direct objects:

• podał łapę = gave a paw
• położył głowę = laid/placed a head

For many feminine nouns ending in -a, the accusative singular changes -a to -ę:

• łapa → łapę
• głowa → głowę

These are past tense masculine singular forms.

The subject is pies, which is grammatically masculine singular, so the verbs must match it:

• pies podał
• pies położył

Compare:

• kot podał = the male cat gave
• suka podała = the female dog gave
• psy podały / podały?
  For non-masculine-personal animal plurals, it is podały.

So the -ł here is part of the masculine singular past tense form.

Because the sentence describes two completed single actions:

1. the dog gave its paw
2. then it put its head on the speaker's knee/lap

Both podał and położył are perfective past forms, which present the actions as whole, finished events.

If you used imperfective verbs, the meaning would change:

• podawał = was giving / used to give / kept offering
• kładł = was laying / kept putting down

So the original sentence sounds like a simple sequence of completed actions, which is exactly what perfective verbs are good for.

Because in Polish, possessive words are often omitted when the owner is obvious, especially with body parts.

So:

• podał mi łapę naturally means gave me his/its paw
• położył głowę naturally means laid his/its head

You could add a possessive word for emphasis:

• podał mi swoją łapę
• położył swoją głowę na moim kolanie

But in this sentence that would sound less natural unless you wanted extra emphasis.

A here links the two actions and gives a smooth narrative flow: and then / and after that.

• a potem = and then, and after that
• i potem = also possible, but it can sound a bit more like simple addition

In storytelling, Polish often uses a to move from one event to the next. It does not always mean a strong contrast; often it just means something like and then.

So:

• Pies podał mi łapę, a potem położył głowę... = very natural narrative style

Moim kolanie is in the locative singular.

The noun is:

• kolano = knee

After the preposition na, Polish can use different cases depending on meaning. Here, the phrase describes the place where the head ended up resting, so locative is used:

• na moim kolanie = on my knee

Also notice agreement:

• moim matches kolanie
• both are singular
• both are in the locative
• kolano is neuter

That is a very common difference between the two languages.

Polish often uses kolano or kolana in contexts where English prefers lap. So the literal meaning is on my knee, but in context it may correspond to English on my lap.

That does not mean the Polish is wrong or strange; it is just a different way of expressing the same situation.

So in this sentence:

• literal Polish image: on my knee
• natural English meaning: often on my lap

No, Polish word order is fairly flexible, but the original order is very natural and neutral.

• Pies podał mi łapę is the normal way to say it
• Mi pies podał łapę could be used for emphasis on to me
• Łapę podał mi pies is possible, but more marked or literary

The same applies to the second half of the sentence.

One important detail: short pronouns like mi often prefer an early position in the clause, so podał mi łapę sounds more natural than podał łapę mi in a neutral sentence.

Yes. That would also be correct.

Polish allows a lot of flexibility with adverbs like potem. But the original version:

• Pies podał mi łapę, a potem położył głowę na moim kolanie

sounds especially smooth because a potem clearly links the second action to the first one.

So your version is grammatical, but the original is a bit more elegant as a short narrative sentence.