Musiałam wyjąć inhalator z plecaka i wziąć tylko mały łyk herbaty.

Polish often leaves out subject pronouns when they are clear from the verb form.

Here, musiałam already tells you I had to, so ja is not necessary.
You could say Ja musiałam..., but that would usually add emphasis, like I had to....

Musiałam is the past tense of musieć (to have to / must).

It tells you three things at once:

• 1st person singular = I
• past tense = had to
• feminine speaker = the speaker is female

A male speaker would say musiałem instead.

After musieć, Polish normally uses the infinitive, just like English uses to + verb after have to.

So:

• musiałam wyjąć = I had to take out
• musiałam wziąć = I had to take

In this sentence, the second infinitive is connected by i:

• Musiałam wyjąć inhalator z plecaka i wziąć tylko mały łyk herbaty.

That means the speaker had to do both actions.

This is about aspect, which is very important in Polish.

• wyjąć and wziąć are perfective
• wyjmować and brać are imperfective

Here the sentence describes single completed actions in a specific situation, so the perfective forms sound natural:

• wyjąć inhalator = take the inhaler out
• wziąć łyk = take a sip

If you used the imperfective forms, it would sound more like an ongoing action, repeated action, or a more general situation.

Because inhalator is the direct object of wyjąć, so it should be in the accusative case.

But inhalator is a masculine inanimate noun, and in the singular, masculine inanimate nouns usually have the same form in nominative and accusative.

So:

• nominative: inhalator
• accusative: inhalator

That is why the form does not change.

The preposition z here means out of / from, and in this meaning it takes the genitive case.

So:

• basic form: plecak = backpack
• after z: z plecaka = out of the backpack / from the backpack

This is a very common pattern in Polish:

• z domu = from the house
• z torby = from the bag
• z plecaka = from the backpack

Because łyk works like sip of..., and the thing after it usually goes in the genitive.

So:

• łyk czego? = a sip of what?
• łyk herbaty = a sip of tea

That is why it is herbaty, not herbatę.

Also note:

• mały łyk is the direct object of wziąć
• herbaty depends on łyk

So the structure is:

• wziąć [co?] mały łyk [czego?] herbaty

Yes. It is a normal, idiomatic way to say take a sip.

So:

• wziąć łyk herbaty = take a sip of tea

Other natural possibilities exist too, depending on style and nuance, for example:

• napić się trochę herbaty = drink a little tea
• łyknąć herbaty = take a gulp / sip some tea

But wziąć łyk is perfectly natural.

Tylko means only / just.

Here it limits the amount: the speaker took only a small sip of tea.

Placing tylko before mały łyk herbaty makes it clear that the sip is what is being limited:

• wziąć tylko mały łyk herbaty = take only a small sip of tea

Polish word order is fairly flexible, but this placement is very natural.

Yes, i means and, but in this sentence it also naturally suggests sequence because of the meaning.

So the listener understands:

1. the speaker had to take the inhaler out of the backpack
2. then take only a small sip of tea

Polish does not need an extra word like then here, because the order is already clear from the context and the logic of the actions.