Deze doos is zwaar; kun jij die tillen?

Because doos is a de-word: de doos.

In Dutch:

• deze = this with de-words
• dit = this with het-words

So:

• deze doos = this box
• but dit boek = this book because boek is a het-word

Here die refers back to deze doos. In natural English, you would usually translate it as it.

Dutch often uses demonstrative words like die and dat to point back to something already mentioned:

• die for de-words
• dat for het-words

Since doos is a de-word, die is correct.

So:

• Deze doos is zwaar; kun jij die tillen? literally feels a bit like:
• This box is heavy; can you lift that one?

But in normal English, can you lift it? is the best translation.

Yes, often you can.

• Kun jij die tillen?
• Kun jij hem tillen?

Both can work. The difference is mainly one of feel:

• die sounds a bit more pointing/emphatic, as if you are indicating that particular box
• hem is a regular object pronoun, more like plain it/him

Because the sentence already starts with deze doos, using die sounds very natural.

Because this is a yes/no question, and in Dutch the finite verb comes first.

Statement:

• Jij kunt die tillen. = You can lift it.

Question:

• Kun jij die tillen? = Can you lift it?

Also, with jij/je, the -t often disappears when the verb comes before the subject:

• jij kunt
• but kun jij?

That is a very common Dutch pattern.

Because kun is a modal verb, and in Dutch the main verb usually goes to the end when a modal verb is present.

So:

• kun = finite/modal verb
• tillen = main verb in the infinitive

Pattern:

• Kun jij die tillen? literally:
• Can you it lift?

This is normal Dutch word order with modal verbs:

• Ik kan zwemmen.
• Wij willen gaan.
• Kun jij die tillen?

Because zwaar is used here after the verb is. That is called a predicative adjective, and in that position Dutch adjectives usually do not take -e.

Compare:

Before the noun:

• de zware doos = the heavy box

After is:

• de doos is zwaar = the box is heavy

So:

• zware before a noun
• zwaar after is

Both mean you, but jij is more stressed or emphatic.

So:

• Kun jij die tillen? can sound like Can you lift it?
• Kun je die tillen? is a little more neutral and casual

In everyday spoken Dutch, je is very common.
Using jij may suggest contrast or emphasis, for example:

• Ik kan het niet. Kun jij die tillen? = I can’t do it. Can you lift it?

Yes. The semicolon links two closely related clauses:

• Deze doos is zwaar; kun jij die tillen?

It shows that the second part follows directly from the first: the box is heavy, so the speaker asks whether you can lift it.

You could also write:

• Deze doos is zwaar. Kun jij die tillen?
• Deze doos is zwaar, kun jij die tillen? (less formal / more conversational punctuation)

The semicolon is just a neat way to connect the two thoughts.

Yes, deze doos is singular: this box.

The plural is:

• deze dozen = these boxes

A useful pattern:

• singular de-word: deze
• plural: deze as well

So:

• deze doos = this box
• deze dozen = these boxes

And for that/those:

• die doos = that box
• die dozen = those boxes