Pielęgniarka zrobiła mi opatrunek na palec.

Because pielęgniarka means a female nurse, and in Polish the past tense agrees with the subject’s gender.

• pielęgniarka zrobiła = a female nurse did it
• pielęgniarz zrobił = a male nurse did it

So the -ła ending in zrobiła shows feminine singular.

Literally, zrobiła can mean made or did, but here it is part of the expression zrobić opatrunek.

In this context, it means something like:

• applied a dressing
• put a bandage on
• dressed the wound

So it is not really about physically manufacturing the dressing.

Mi is the dative form of ja and means to me or for me.

In natural English, though, you would often translate it as part of my finger:

• zrobiła mi opatrunek na palec
  = she put a dressing on my finger

Polish often uses a dative pronoun like mi with body parts instead of a possessive like my.

Because Polish usually does not repeat possession when it is obvious, especially with body parts.

So instead of saying something equivalent to on my finger, Polish often says something more like to me on the finger.

You could say na mój palec, but that would sound more explicit and is usually unnecessary unless you want contrast or special emphasis.

Because opatrunek is the direct object of zrobiła, so it is in the accusative case.

The important detail is that opatrunek is a masculine inanimate noun, and in the singular its accusative looks the same as its nominative.

So:

• nominative: opatrunek
• accusative: opatrunek

That is why the form does not visibly change.

This is a very common Polish pattern.

With na, Polish uses:

• accusative for movement or placement onto something
• locative for location on something

So:

• na palec = onto the finger / applied to the finger
• na palcu = on the finger, as a location or state

Compare:

• Pielęgniarka zrobiła mi opatrunek na palec.
  The action is putting it onto the finger.

• Mam opatrunek na palcu.
  The dressing is already on the finger.

It is accusative singular, because it follows na in a sense of placement.

For this noun, the accusative singular looks the same as the nominative:

• nominative: palec
• accusative: palec
• locative: palcu

So the contrast is really:

• na palec = accusative
• na palcu = locative

Zrobiła is perfective.

That means the action is seen as completed: the nurse finished putting on the dressing.

If you used robiła, that would be imperfective, suggesting:

• she was doing it
• she used to do it
• the action is viewed as ongoing or repeated

In this sentence, zrobiła fits because it refers to one completed medical action.

No, Polish word order is fairly flexible, but this order is natural and neutral:

Pielęgniarka zrobiła mi opatrunek na palec.

The small word mi is an unstressed pronoun, so it usually stays near the beginning of the clause, often after the verb or after the first stressed element.

Changing the order is possible, but it usually changes the emphasis rather than the basic meaning.

Polish has no articles, so there are no direct equivalents of a and the in sentences like this.

Whether something is a nurse or the nurse, a dressing or the dressing, is understood from context.

Yes, it is understandable and natural. But Polish also has other common ways to say something similar, for example:

• Pielęgniarka założyła mi opatrunek na palec.
• Pielęgniarka opatrzyła mi palec.

These can sound a bit more specific than zrobiła, but the original sentence is perfectly good Polish.