Weterynarz kazał mi pilnować psa przez dwa dni i nie biegać z nim za długo.

This is a very common Polish pattern:

kazać komuś coś zrobić

It means to order / tell someone to do something.

So here:

• Weterynarz = the vet
• kazał = ordered / told
• mi = me
• pilnować... i nie biegać... = to watch over... and not run...

In other words, both infinitives depend on kazał.

Both mean me in the dative, but they are used differently.

• mi = the normal short, unstressed form
• mnie = the fuller form, often used for emphasis, contrast, or after some prepositions

Here the sentence is neutral, so mi is the natural choice:

• Weterynarz kazał mi...

If you said mnie, it would sound more emphatic, like the vet told me.

Because pies is the basic dictionary form, but after pilnować you use another case form.

The verb pilnować traditionally takes the genitive:

• pilnować kogo? czego?

For pies, the genitive singular is psa.

So:

• pies = dog
• pilnować psa = to watch/keep an eye on the dog

A useful thing to notice: for masculine animate nouns, the genitive and accusative often look the same, so psa is a very common form to see.

Pilnować means something like:

• to watch over
• to keep an eye on
• to supervise
• to make sure someone/something is okay and behaves properly

In this sentence, it does not mean staring at the dog every second. It suggests looking after the dog carefully, especially because the instruction comes from a vet.

Przez + accusative is a common way to express duration, especially for a period of time.

So:

• przez dwa dni = for two days

Here dwa dni is in the accusative after przez.

This is a very useful pattern:

• przez godzinę = for an hour
• przez tydzień = for a week
• przez dwa dni = for two days

Sometimes Polish can express duration without przez, but in this sentence przez dwa dni is very clear and natural.

Using przez makes the time span explicit:

• for two days

So for a learner, przez + time expression is a safe and useful pattern to remember.

This is about the type of motion verb.

• biec = to run in a single instance, usually in one direction
• biegać = to run around, to go running, to run repeatedly/habitually, or as a general activity

Here the vet is talking about a general activity the dog should not do for too long, so biegać sounds natural.

Compare:

• Pies biegnie do domu. = The dog is running home.
• Pies lubi biegać. = The dog likes running.

In this sentence, it is the second kind of meaning.

Because z meaning with takes the instrumental case.

The pronoun on changes like this:

• on = he
• nim = him (after z, in the instrumental form)

So:

• z nim = with him

You cannot say z on or z go.

Za before adjectives and adverbs often means too.

So:

• długo = long / for a long time
• za długo = too long

Very common similar examples:

• za szybko = too fast
• za późno = too late
• za dużo = too much

So nie biegać z nim za długo means not to run with him for too long.

Yes, it does. Both infinitives belong to the same verb kazał.

The structure is basically:

• Weterynarz kazał mi [pilnować psa przez dwa dni] i [nie biegać z nim za długo].

Polish does not need to repeat mi here, because it is understood for both actions.

Repeating it would usually sound unnecessary.

Yes, but it would be a different structure.

For example:

• Weterynarz kazał, żebym pilnował psa przez dwa dni i nie biegał z nim za długo.

That is possible, but in this kind of sentence the infinitive structure with kazać is very natural and compact:

• kazał mi pilnować...
• kazał mi nie biegać...

So the version with infinitives is a very standard way to say it.

In the past tense, Polish verbs agree with the gender of the subject.

So:

• kazał = masculine singular
• kazała = feminine singular

This sentence shows a male vet:

• Weterynarz kazał mi...

If the vet were female, you would say:

• Weterynarz kazała mi...

Even if the profession noun is still weterynarz, the verb can show that the person is female.

Not completely. Polish word order is fairly flexible.

This version is natural and clear:

• Weterynarz kazał mi pilnować psa przez dwa dni i nie biegać z nim za długo.

But other orders are possible, for example:

• Weterynarz kazał mi przez dwa dni pilnować psa...

That said, changing the order can slightly change emphasis. In the original sentence:

• przez dwa dni most naturally goes with pilnować psa
• za długo clearly goes with biegać z nim

So the original order is a good neutral choice.

In this sentence, it most naturally applies to pilnować psa.

So the structure is understood as:

• watch the dog for two days
• and do not run with him for too long

The second part already has its own time-related phrase: za długo.

If you wanted for two days to apply equally to both actions, Polish would usually rephrase the sentence to make that clearer.

Because in Polish, nie is generally written separately from verbs.

So:

• nie biegać
• nie robić
• nie wiem

That is the normal spelling rule for verbs.

This is useful to remember because with some other word types, Polish may write nie together, but with verbs it is usually separate.