Masz czym otworzyć tę puszkę z tuńczykiem?

Because Polish often drops subject pronouns when they are clear from the verb form.

Masz already means you have.
So Masz czym otworzyć tę puszkę z tuńczykiem? naturally means Do you have something to open this tuna can with?

You could add ty for emphasis:

Ty masz czym otworzyć tę puszkę z tuńczykiem?

But that would sound more marked, like You have something to open it with?

Polish does not use a helper verb like English do to make yes/no questions.

A sentence can become a question simply through:

• intonation
• context
• sometimes the particle czy

So all of these can work:

• Masz czym otworzyć tę puszkę z tuńczykiem?
• Czy masz czym otworzyć tę puszkę z tuńczykiem?

Both mean the same thing. The version without czy is very common in everyday speech.

Czym is the instrumental case form of co. In this kind of sentence, Polish uses the pattern:

mieć czym + infinitive

This means something like:

• to have something with which to do X
• to have a way / tool / means to do X

So:

• Masz czym otworzyć... = Do you have something to open ... with?

If you used coś, you would usually need a fuller structure, for example:

Masz coś, czym można otworzyć tę puszkę?
= Do you have something that one can open this can with?

That is grammatical, but less compact.

It is a very common Polish structure. Literally, it is close to:

You have with-what open...

But idiomatically it means:

• Do you have anything to open it with?
• Do you have a way to open it?

This pattern is used with many verbs:

• Nie mam czym pisać. = I don’t have anything to write with.
• Masz czym zapłacić? = Do you have anything to pay with?
• Nie było czym oddychać. = There was nothing to breathe. / There was no air to breathe.
  (more literally: there was no way to breathe)

So czym often points to a tool, means, or available resource.

Because otworzyć is perfective, and here the speaker means one complete action: opening this specific can.

• otworzyć = to open completely / to get it open
• otwierać = to be opening / to open repeatedly / in general

In this sentence, the question is about whether you have a way to achieve the result once.

So:

• Masz czym otworzyć tę puszkę? = Do you have something to open this can with?

If you said:

• Masz czym otwierać puszki?

that would sound more like:

• Do you have something for opening cans?
• Do you have a tool you use to open cans in general?

Because otworzyć takes a direct object, and here that object is in the accusative case.

The noun puszka is feminine, so in the accusative singular:

• puszka → puszkę

The demonstrative ta also changes:

• ta → tę

So:

• ta puszka = this can
• otworzyć tę puszkę = to open this can

A useful note: in everyday speech, many native speakers say tą puszkę, but in standard written Polish, tę puszkę is preferred.

Because z in the sense of with takes the instrumental case, so:

• tuńczyk → tuńczykiem

So puszka z tuńczykiem literally means:

• a can with tuna
• more naturally, a can containing tuna

This is a very normal way to describe what is inside a container.

You may also hear:

• puszka tuńczyka = a can of tuna

That is also natural, but the structure is different:

• puszka z tuńczykiem focuses on a can with tuna in it
• puszka tuńczyka is closer to English a can of tuna

In this sentence, puszkę z tuńczykiem is perfectly idiomatic.

No. Czym does not name a specific tool. It just means something / anything to do it with.

That could be:

• a can opener
• a knife
• scissors
• some other object

So the question is broader than:

• Masz otwieracz do puszek? = Do you have a can opener?

Instead, it means:

• Do you have anything we can use to open this can?

Yes, Polish word order is fairly flexible, but not all versions sound equally natural.

The most natural order here is:

Masz czym otworzyć tę puszkę z tuńczykiem?

That keeps czym otworzyć together as a natural unit.

You might also hear:

• Czy masz czym otworzyć tę puszkę z tuńczykiem?

Other word orders are possible for emphasis, but they may sound less neutral or less smooth.

So for a learner, this sentence is a very good standard pattern to remember as-is.

Some natural answers would be:

• Tak. = Yes.
• Tak, mam. = Yes, I do.
• Tak, mam otwieracz. = Yes, I have a can opener.
• Tak, mam nóż. = Yes, I have a knife.
• Nie, nie mam czym. = No, I don’t have anything to open it with.
• Nie, ale możemy spróbować nożem. = No, but we can try with a knife.

Notice again the useful pattern:

• nie mam czym = I don’t have anything to do it with