Ufam tej lekarce, a jeśli chcesz, żeby dziecko ufało opiekunce, musi ją dobrze poznać.

Because the verb ufać requires the dative case, not the accusative.

• dictionary form: lekarka, opiekunka
• after ufać:
  
  • ufać lekarce = to trust a female doctor
  • ufać opiekunce = to trust a female caregiver

So:

• Ufam tej lekarce = I trust this female doctor
• żeby dziecko ufało opiekunce = so that the child would trust the caregiver

This is one of the most important things to remember with ufać: it goes with komu? czemu? (dative), not with direct object forms.

For the same reason: ufać takes the dative.

The phrase ta lekarka changes like this:

• nominative: ta lekarka
• accusative: tę lekarkę
• dative: tej lekarce

Since the sentence uses ufać, you need the dative:

• Ufam tej lekarce

If the verb took a direct object, you might see tę lekarkę, but not here.

This is a very common learner question. After żeby, Polish often uses a form that looks like the past tense, but its meaning is not actually past here.

So:

• żeby dziecko ufało opiekunce means so that the child would trust the caregiver / for the child to trust the caregiver

The form ufało is used because:

• the subject is dziecko,
• dziecko is grammatically neuter singular,
• so the żeby clause uses the neuter singular past-style form: ufało

Compare:

• żeby chłopiec ufał = so that the boy would trust
• żeby dziewczynka ufała = so that the girl would trust
• żeby dziecko ufało = so that the child would trust

So yes, it looks like past tense, but after żeby it often has this subjunctive-like / desired-result meaning.

Musi refers to dziecko.

The structure is:

• jeśli chcesz, żeby dziecko ufało opiekunce, musi ją dobrze poznać

The understood subject of musi is the same child:

• the child must get to know her well

In Polish, subject pronouns are often omitted when they are clear from context. So instead of saying:

• ono musi ją dobrze poznać

Polish usually just says:

• musi ją dobrze poznać

because dziecko has already been mentioned.

Ją is the accusative form of ona and refers to opiekunka.

In:

• musi ją dobrze poznać

the verb poznać takes a direct object, so you need the accusative:

• poznać kogo? co?
• poznać opiekunkę
• pronoun: ją = her

So:

• ufać opiekunce → dative
• poznać ją → accusative

This contrast is very useful:

• ufać komuś = trust someone
• poznać kogoś = get to know / meet someone

Because ją is the normal accusative form used without a preposition.

• Widzę ją. = I see her.
• Musi ją dobrze poznać. = It/he/she must get to know her well.

Nią is typically used after a preposition or in instrumental contexts, for example:

• z nią = with her
• nad nią = over her

So in this sentence, ją is the correct choice.

This is a very important distinction.

• znać = to know, be familiar with
• poznać = to get to know, come to know, meet

So:

• znać opiekunkę = to know the caregiver
• poznać opiekunkę = to get to know the caregiver

In this sentence, poznać is used because the child needs to become familiar with the caregiver first. That fits the idea:

• first you poznajesz someone,
• then you may znasz them,
• and maybe eventually ufasz them.

Also, poznać is perfective, so it suggests reaching that result: getting to know her successfully.

They are two different forms of the same verb ufać.

• ufam = I trust
• ufało = would trust / trusted, neuter singular form, here used after żeby

So:

• Ufam tej lekarce = I trust this doctor
• żeby dziecko ufało opiekunce = so that the child would trust the caregiver

This is a nice example of how Polish changes the verb depending on:

• person: ufam = first person singular
• clause type: żeby ... ufało
• gender/number of the subject: dziecko is neuter singular

Because Polish often drops subject pronouns when the verb form already shows who the subject is.

Examples from the sentence:

• ufam already means I trust
• chcesz already means you want
• musi can stand without a pronoun because the subject is understood from context: dziecko

So Polish commonly says:

• Ufam tej lekarce not
• Ja ufam tej lekarce

And:

• jeśli chcesz... not necessarily
• jeśli ty chcesz...

The pronouns ja, ty, ono are usually added only for emphasis, contrast, or clarity.

Here żeby introduces a clause expressing a desired result or intention.

In:

• jeśli chcesz, żeby dziecko ufało opiekunce...

it means something like:

• if you want the child to trust the caregiver
• if you want the child to be trusting toward the caregiver

A very common pattern is:

• chcieć, żeby... = to want someone/something to...

Examples:

• Chcę, żeby przyszedł. = I want him to come.
• Chcesz, żeby dziecko spało. = You want the child to sleep.
• Chcesz, żeby dziecko ufało opiekunce. = You want the child to trust the caregiver.

A often links two clauses and can mean something between and, while, or whereas, depending on context.

Here:

• Ufam tej lekarce, a jeśli chcesz...

it works like a smooth connector: and / and as for / while

It does not necessarily mean a strong but here. It simply moves from one idea to the next:

• I trust this doctor, and if you want the child to trust the caregiver, ...

Polish a is often softer and more flexible than English learners expect.

Because dziecko is grammatically a neuter noun in Polish.

That affects agreement:

• dziecko ufało
• dziecko było
• małe dziecko

So after żeby, you get:

• żeby dziecko ufało

not:

• żeby dziecko ufał
• żeby dziecko ufała

Even though a child is a person, the noun dziecko is grammatically neuter, and Polish grammar follows the noun’s grammatical gender.

Not naturally in this context.

• ufać komuś = to trust someone
• wierzyć komuś = to believe someone
• wierzyć w coś / w kogoś = to believe in something / someone

So:

• Ufam tej lekarce = I trust this doctor
• Wierzę tej lekarce = I believe this doctor

Those are related ideas, but not the same. In the sentence you gave, the idea is about trust, so ufać is the correct verb.

Yes, the commas are normal and correct in Polish.

You have commas because:

1. a links two clauses:

   • Ufam tej lekarce, a ...

2. jeśli introduces a subordinate clause:

   • a jeśli chcesz, ...

3. żeby introduces another subordinate clause:

   • jeśli chcesz, żeby dziecko ufało opiekunce, ...

So the sentence is built from several parts, and Polish punctuation marks those boundaries clearly.

A rough structure is:

• Ufam tej lekarce,
• a jeśli chcesz,
• żeby dziecko ufało opiekunce,
• musi ją dobrze poznać.

That is why the commas are there.