Remélem, hogy a tápkábel és a vasaló is a dobozban van.

Because Hungarian often uses the definite conjugation before a hogy-clause, as if there were an understood object like azt = that.

So Remélem, hogy... is basically short for (Azt) remélem, hogy... = I hope that...

• remélek = I hope in a more general/indefinite sense
• remélem = I hope it / I hope that...

With a following clause, remélem is the normal form.

Hungarian usually does not need subject pronouns, because the verb ending already shows the person and number.

So Remélem already means I hope.
Adding én is possible, but it usually adds emphasis or contrast:

• Én remélem... = I hope..., as opposed to someone else

Without emphasis, leaving out én is completely normal.

Hogy introduces a subordinate clause and usually corresponds to English that.

So:

• Remélem, hogy... = I hope that...

In English, that is often optional. In Hungarian, hogy is very common and natural in this structure.

Because in standard Hungarian spelling, a subordinate clause introduced by hogy is normally separated by a comma.

So Remélem, hogy... is the standard written form.

A is the definite article, like English the.

Here both nouns are definite, so Hungarian naturally says:

a tápkábel és a vasaló

Repeating the article before both nouns is very normal when talking about two specific things. It makes the coordination clear and natural.

It is a compound noun:

• táp = power supply / power
• kábel = cable

So tápkábel literally means something like power cable, which in natural English may also be power cord.

Hungarian uses compound words very freely, much more than English does.

Yes, and that is exactly the idea.

• és = and
• is = also / too / as well

In a tápkábel és a vasaló is, the is adds the sense of also to the coordinated phrase. In natural English, that often comes out as:

• the power cable and the iron are also in the box
• the power cable and the iron too are in the box

So és links the two nouns, while is adds the meaning also.

In Hungarian, is usually comes after the word or phrase it belongs to.

Here it appears after a vasaló, but in this kind of coordination it can be understood as applying to the whole phrase a tápkábel és a vasaló.

That is why the sentence means not just the cable and the iron are in the box, but the cable and the iron are also in the box.

If you removed is, the extra also/too meaning would disappear.

A dobozban means in the box.

• a doboz = the box
• -ban = in

This ending is called the inessive case, used for being inside something.

So:

• doboz = box
• dobozban = in a/the box

Because of vowel harmony.

Hungarian suffixes often change depending on the vowels in the word:

• back-vowel words usually take -ban
• front-vowel words usually take -ben

Doboz contains back vowels, so it becomes dobozban.

Compare:

• házban = in the house
• szekrényben = in the cupboard

In a neutral Hungarian sentence of this type, the location often comes before van.

So Hungarian prefers:

• a dobozban van = is in the box

This is normal word order for saying where something is.

English says is in the box, but Hungarian often places the location before the verb:

• A könyv az asztalon van. = The book is on the table.

That is a very good question, because many learners would indeed expect vannak.

In careful standard Hungarian, with two separate items as the subject, vannak is often the safest textbook form:

Remélem, hogy a tápkábel és a vasaló is a dobozban vannak.

However, singular van can also occur in everyday speech, especially in list-like or packaging contexts, where the speaker is thinking of the contents as a set or mentioning them somewhat collectively.

So:

• vannak = the safer, more standard choice for two items
• van = something you may still hear in real usage

If you want the most unambiguously standard version, use vannak.

Yes, especially in casual speech.

You can say:

Remélem, a tápkábel és a vasaló is a dobozban van / vannak.

That sounds a bit more conversational.
With hogy, the sentence is slightly more explicit and is also completely natural.

So both are possible, but Remélem, hogy... is a very common standard pattern.