Ons moet die tafel afvee sonder om die water te mors.

Because moet is a modal verb, and in Afrikaans the modal is the finite verb in the main clause. The finite verb normally goes in second position, while the other verb goes to the end.

So:

• Ons = subject
• moet = finite verb, in second position
• die tafel = object
• afvee = main verb, at the end

That is why you get:

Ons moet die tafel afvee.

This is very similar to Dutch and German word order.

Afvee is a separable verb:

• vee = wipe / sweep
• af = off

In some sentence patterns it splits, and in others it stays together.

It stays together here because it is being used as an infinitive after moet:

• Ons moet die tafel afvee.

But in a normal present-tense clause, it splits:

• Ons vee die tafel af.

And with te, the te goes between the parts:

• om die tafel af te vee

So all of these are forms of the same verb:

• afvee
• vee ... af
• af te vee

This is the standard Afrikaans way to say without doing something.

So:

• sonder = without
• om ... te + verb = infinitive structure

In this sentence:

sonder om die water te mors

means:

without spilling the water

A useful rule is:

• sonder + noun → no om
  • sonder water = without water
• sonder + verb idea → usually sonder om ... te
  • sonder om te praat = without speaking

Because the two verbs belong to different structures.

After a modal verb like moet, Afrikaans uses a bare infinitive:

• moet afvee
• kan lees
• wil gaan

So you do not say moet te afvee.

But the structure om ... te requires te:

• om te mors
• om te lees
• om te gaan

So in this sentence:

• moet afvee = modal + bare infinitive
• om ... te mors = infinitive clause with om ... te

Because in Afrikaans, the object usually comes before the infinitive verb at the end of the clause.

So the pattern is often:

• om + object + te + verb

That gives:

• om die water te mors

not:

• om te mors die water

This same end-of-clause tendency is very common in Afrikaans verb structures.

Here, mors means spill.

But mors is a flexible word and can also mean:

• make a mess
• waste

Examples:

• Moenie die melk mors nie. = Don’t spill the milk.
• Moenie jou tyd mors nie. = Don’t waste your time.

In your sentence, because the object is die water, the natural meaning is spill the water.

No—modern Afrikaans does not use different definite articles for grammatical gender the way Dutch or German does.

Die is the definite article for basically all nouns:

• die tafel = the table
• die water = the water
• die man = the man
• die vrou = the woman
• die kinders = the children

So you do not need to learn separate words for the based on gender.

It can mean either, depending on context.

So:

Ons moet die tafel afvee

can be understood as:

• We must wipe off the table
• We have to wipe off the table

In everyday use, moet often covers both ideas of obligation and necessity. The exact English translation depends on how strong the obligation feels in context.

Yes, ons can mean different things depending on its role in the sentence:

• we
• us
• our

Here it means we because it is the subject of the sentence and comes before the finite verb moet:

• Ons moet ... = We must ...

Compare:

• Hy sien ons. = He sees us.
• ons tafel = our table

So the grammar of the sentence tells you which meaning ons has.

You would normally say:

Ons vee die tafel af sonder om die water te mors.

This is useful because it shows the separable verb in its split form:

• vee ... af

So comparing the two helps a lot:

• Ons moet die tafel afvee.
• Ons vee die tafel af.

That is one of the most important verb patterns to get used to in Afrikaans.