Learn English – “demand” vs “insist”

According to the definition, insist by itself does not make anything happen.

So you should use should since it is natural to consider doing what your mother says, but you certainly don't have to. (In this statement I also use "should", since it is my recommendation but ultimately your choice how to say it.)

So it is correct to say:

My mother insisted that I should stay at home for a couple of days due to my illness.

am is better suited to a statement where something will certainly be done. For example:

I don't care what my mother says, I am going to that party.

The form "insist on" is used with nouns and noun phrases; whereas "insist" alone is used to with finite clauses (clauses that can be sentences on their own).

For the three examples that you gave, two are grammatical:

• I insist that he does his homework now.
• I insist he does his homework now.

These are both okay, because "that" is usually optional when used to introduce a finite relative clause.

The other example is not:

• I insist on that he does his homework now.

This is not grammatical because "that he does his homework now" is not a noun phrase. As in Tᴚoɯɐuo's example, you could turn the latter part of the sentence into a noun phrase with "I insist on him doing his homework now".