Learn Hardcore Dutch: Kun jij vanavond op de kinderen oppassen, of zal ik het doen? - Can you babysit the children tonight, or shall I do it?

With inversion (verb before subject), the -t drops: jij kunt → kun jij?
Modern Dutch also allows jij kan, so kan jij? is common too.
Kun je is equally correct and a bit more neutral/colloquial.
jij (full form) is used when the speaker stresses you in particular; je is the unstressed form. So Kun jij… puts a bit of emphasis on you.

Why is oppassen at the very end?

Because kun is a modal verb. In main clauses:

The finite verb goes to the second position: Kun.
The main lexical verb appears as a bare infinitive at the end: oppassen. Compare:
Without a modal: Pas jij vanavond op de kinderen?
With a modal: Kun jij vanavond op de kinderen oppassen?

Why do we need op? Why not just de kinderen oppassen?

The meaning “to look after/babysit” is expressed by the combination (op)passen op + object. The object is introduced by the preposition op:

Correct: op de kinderen passen / op de kinderen oppassen
Not idiomatic: de kinderen oppassen (without op). Without op, passen means something else (fit/try on, be suitable).

Which is more idiomatic: op de kinderen oppassen or op de kinderen passen?

Both are used and understood. Many speakers (and some style guides) treat op de kinderen passen as the neutral default when the object is expressed; op de kinderen oppassen is very common in everyday speech and not wrong. If you want a “safest” choice in careful writing, use op de kinderen passen.

What exactly is going on in op de kinderen oppassen? Isn’t op there twice?

They are two different things that just look the same:

op before de kinderen is the preposition that introduces the object.
op- inside oppassen is the separable verb particle. With a modal, the infinitive oppassen stays together at the end, so both appear:
Ik kan op de kinderen oppassen. Without a modal, you see only the preposition:
Ik pas op de kinderen.

Can I also say Kun jij vanavond oppassen op de kinderen?

You will hear this, especially in colloquial Dutch and in Belgium. It’s understandable, but in neutral standard Dutch it’s more idiomatic to keep the prepositional phrase before the infinitive: Kun jij vanavond op de kinderen oppassen? Alternatively: Kun jij vanavond op de kinderen passen?

Why is there a comma before of?

of links two main clauses here. A comma before coordinating of is optional in Dutch; many writers insert it to make the pause clear. So both are fine:

Kun jij …, of zal ik …?
Kun jij … of zal ik …?

What does het refer to in zal ik het doen, and can I omit it?

het is a pro-form that stands for the whole action “babysitting the children.” In conversation you can often omit it or replace the whole thing with the full activity:

Zal ik (het) doen?
Zal ik (vanavond) op de kinderen (op)passen?
Very colloquial: Zal ik? In careful writing, keep het or repeat the activity for clarity.

Why use zal here—does it mean future tense?

In Dutch, zullen with the first person is commonly used to make an offer or suggestion: Zal ik …? ≈ “Shall I …?” It’s not about future time here; it’s about proposing to do something.

Is Kun jij …? polite enough? Would Wil je …? or Zou je (kunnen/willen) …? be better?

All are possible, with slightly different nuance:

Kun je/jij …? neutral request (ability as a politeness strategy).
Wil je/jij …? asks about willingness; also very common.
Zou je (kunnen/willen) …? more tentative/polite. Pick based on how deferential you want to sound.

Why isn’t there te before oppassen (as in te oppassen)?

Modal verbs (like kunnen, willen, mogen, moeten, zullen) and laten take a bare infinitive in Dutch. So: Kun jij … oppassen, not te oppassen.

Can I put vanavond somewhere else in the sentence?

Yes. Word order is flexible for adverbs:

Neutral: Kun jij vanavond op de kinderen oppassen …
With fronting/emphasis: Vanavond, kun jij op de kinderen oppassen …
End position (a bit more emphatic): Kun jij op de kinderen oppassen vanavond … All are grammatical; choose based on what you want to emphasize.

Why use jij and not je here?

jij is the stressed form and highlights the subject. Kun jij … subtly contrasts you with someone else (“can you (rather than me/another person)?”). Kun je … is the unstressed, more neutral variant.

How would this look with the formal u?

Kunt u vanavond op de kinderen passen, of zal ik het doen?
Note the form kunt with u.

What’s the difference between Oppassen!/Pas op! and op de kinderen oppassen?

Oppassen!/Pas op! = “Watch out!” (general warning).
(Op) de kinderen (op)passen = “to look after/babysit the children.” Same verb family, different usages.

Can I replace op de kinderen with a pronoun?

Yes:

If they’re already known in context: Kun je er vanavond op passen? (pronominal adverb: er … op)
For people, many speakers also say: Kun je vanavond op ze passen? Both are idiomatic. Use er … op more for things; op ze is natural for people.