Ek weet dat sy eerlik is.

Because dat introduces a subordinate clause. In standard Afrikaans, the finite verb usually goes to the end of a dat-clause.

So:

• Main clause: Sy is eerlik.
• Subordinate clause: dat sy eerlik is

That is why the full sentence is Ek weet dat sy eerlik is.

Here dat is a conjunction meaning that. It links the main clause Ek weet to the statement being known: sy eerlik is.

So the structure is:

• Ek weet = main clause
• dat sy eerlik is = content clause introduced by dat

Yes, very often. A natural alternative is:

Ek weet sy is eerlik.

When you leave out dat, the second clause usually goes back to normal main-clause order:

• met dat: sy eerlik is
• sonder dat: sy is eerlik

Both are common, but the version with dat sounds a little more explicit or formal.

Afrikaans splits English know into two common verbs:

• weet = to know a fact, piece of information, or truth
• ken = to know a person, place, or something through familiarity

In this sentence, the speaker knows a fact: that she is honest. So weet is the correct verb.

Compare:

• Ek weet dat sy eerlik is.
• Ek ken haar.

In Afrikaans, sy can mean either she or his, depending on how it is used.

Here it means she, because it is the subject of the clause:

dat sy eerlik is = that she is honest

If sy meant his, it would normally come before a noun:

• sy huis = his house
• sy vriend = his friend

So the grammar makes the meaning clear.

Because eerlik is being used as a predicate adjective, after the verb is.

In Afrikaans:

• After is: Sy is eerlik.
• Before a noun: 'n eerlike meisie

So:

• eerlik = predicate use
• eerlike = attributive use before a noun

That is why dat sy eerlik is is correct.

Usually no. Afrikaans verbs change much less than English verbs.

You say:

• Ek weet
• Jy weet
• Hy weet
• Ons weet
• Hulle weet

So weet stays the same across subjects in the present tense.

There are two different possibilities, depending on what you want to negate.

If you mean I do not know that she is honest, you can say:

Ek weet nie dat sy eerlik is nie.

If you mean I know that she is not honest, you can say:

Ek weet dat sy nie eerlik is nie.

This shows the normal Afrikaans double negative pattern with nie ... nie.