Zonder flesopener krijg ik de dop niet van de fles, en zonder blikopener lukt het ook niet.

After zonder, Dutch very often uses a bare singular noun when the meaning is general: without a bottle opener, without a can opener.

So:

• zonder flesopener = without a bottle opener / without any bottle opener
• zonder blikopener = without a can opener / without any can opener

You can also say zonder een flesopener, but that is a bit more explicit and often sounds less neutral in a general statement.

Because Zonder flesopener is placed first in the sentence.

Dutch main clauses normally follow the V2 rule: the finite verb comes in the second position. If something other than the subject comes first, the verb still stays second, and the subject moves after it.

So:

• Ik krijg de dop niet van de fles.
• Zonder flesopener krijg ik de dop niet van de fles.

The same thing happens in the second clause:

• Zonder blikopener lukt het ook niet.

Dop usually means a cap, especially on a bottle or tube.

Examples:

• een flessendop = a bottle cap
• de dop van de fles = the cap of the bottle

Deksel is more like a lid, for example on a pan, jar, or box.

So for a bottle, dop is the natural choice.

That is very normal Dutch.

Literally, de dop van de fles krijgen is something like to get the cap from/off the bottle. In natural English, you would usually say to get the cap off the bottle.

Dutch often uses van where English uses off in this kind of removal idea.

So:

• Ik krijg de dop van de fles.
  = I get the cap off the bottle.

With niet:

• Ik krijg de dop niet van de fles.
  = I can’t get the cap off the bottle.

Yes, but krijgen also has a very common extended use: to manage to get something into a certain state.

For example:

• Ik krijg de deur niet open. = I can’t get the door open.
• Ik krijg het etiket er niet af. = I can’t get the label off.
• Ik krijg de dop niet van de fles. = I can’t get the cap off the bottle.

So here krijgen is not about receiving something. It is about successfully managing to remove or open something.

In Dutch, niet usually comes after the direct object and before the final part of the predicate, such as a prepositional phrase or result phrase.

That is why you get:

• Ik krijg de dop niet van de fles.

Here:

• de dop = direct object
• niet = negation
• van de fles = the phrase that completes the idea of removing it

So this word order is the normal, idiomatic one.

Yes. Dutch makes compound nouns very freely.

• fles + opener = flesopener = bottle opener
• blik + opener = blikopener = can opener

In Dutch compounds, the last part is the main word. Here that is opener, so both words mean a kind of opener.

Also, because opener is a de-word, both compounds are de words too:

• de flesopener
• de blikopener

Because lukken works differently from English to succeed or to manage.

Dutch does not normally say ik lukt. Instead, lukken is used in a pattern like:

• Het lukt me. = I manage / It works for me.
• Het lukt me niet. = I can’t manage / It doesn’t work for me.

So in lukt het ook niet, the subject is het, not ik.

A more fully spelled-out version would be:

• Zonder blikopener lukt het me ook niet.

That means Without a can opener, I can’t manage it either or it doesn’t work for me either.

It refers to the action or task that is understood from the context: opening it, getting it off, making it work, and so on.

With lukken, Dutch very often uses het as a kind of general placeholder:

• Het lukt niet. = It doesn’t work / I can’t manage it.
• Het lukt wel. = It works / I can manage it.

So het does not have to refer to one very concrete noun. It can refer to the whole action being discussed.

It is simply left out because the context already makes it clear who is involved.

Full version:

• Zonder blikopener lukt het me ook niet.

Shorter version:

• Zonder blikopener lukt het ook niet.

The shorter version sounds a bit more impersonal, like that also doesn’t work. The speaker is still understood from the context.

Because Dutch uses ook in places where English often uses either in a negative sentence.

So:

• ook = also / too
• ook niet = also not = not ... either

Examples:

• Ik wil ook niet. = I don’t want to either.
• Dat lukt ook niet. = That doesn’t work either.

So in your sentence:

• zonder blikopener lukt het ook niet
  = without a can opener, it doesn’t work either / I can’t manage it either