The question does not make much sense and I think you've gone about as far as anyone can go.

A) at least one character changes

Not necessarily! In any tree, of interest, it will be true that something will have changed but in theory, the only thing that separate two nodes is time. It is not impossible that your teacher was hoping for A to be the correct answer but it is not IMO. It is also very much possible that the only change in between two nodes relate to synonymous mutations (and other mutations that has no effect on the phenotype), hence not causing a change in any character fo interest.

B) all organisms die

Well, most likely enough time separate the two nodes that all organisms that were represented in the ancestral node are dead but of course they have reproduced, so that we actually have another node that descend from it.

C) Nothing can happen between two nodes

As you said, they evolve. So, things happen!

D) Monophyly, Paraphyly, Polyphyly

As you said, those terms make no sense here!

E) None of the above

If there is any intuitive answer to the question what has happened between two nodes is time has passed. I think the answer your teacher was expecting here is E. But really that's a pretty poorly written question IMO.

The question does not make much sense and I think you've gone about as far as anyone can go.

A) at least one character changes

It depends what it means by character and it depends the detail semantic of the tree. In general, an evolutionary tree, usually called a phylogenetic tree attempts at representing time by the length of the edges in between two nodes. The goal is, in general, not to represent anything else.

In the comments, @John has argued that time can be consider as a character. Personally, when I read the term "character" I think of a phenotypic character. While it will pretty much always be true that among two nodes some difference will exist in the phenotypic character, it is not a fundamental consequence of the concept of evolutionary tree. This is why I argued that A is false. @John takes the opposite stance in his answer. Note also, that sometimes (although quite rarely today) phenotypic states are used to build a tree and hence, in such case, it would be a direct consequence of the methodology that two nodes must necessarily differ by the value of at least one phenotypic character.

This disagreement between @John and I highlights that the question is poorly phrased and actually makes little sense.

B) all organisms die

Well, most likely enough time separate the two nodes that all organisms that were represented in the ancestral node are dead but of course they have reproduced, so that we actually have another node that descend from it.

C) Nothing can happen between two nodes

As you said, they evolve. So, things happen!

D) Monophyly, Paraphyly, Polyphyly

As you said, those terms make no sense here!

E) None of the above

If there is any intuitive answer to the question what has happened between two nodes is time has passed. I think the answer your teacher was expecting here is E but it is also possible (s)he was expecting A. Really that's a pretty poorly written question.

The question does not make much sense and I think you've gone about as far as anyone can go.

A) at least one character changes

Not necessarily! Although, very likely at least one new mutation would fix.

B) all organisms die

Well, most likely enough time separate the two nodes that all organisms that were represented in the ancestral node are dead but of course they have reproduced, so that we actually have another node that descend from it.

C) Nothing can happen between two nodes

As you said, they evolve. So, things happen!

D) Monophyly, Paraphyly, Polyphyly

As you said, those terms make no sense here!

E) None of the above

If there is any intuitive answer to the question what has happened between two nodes is time has passed. I think the answer your teacher was expecting here is E. But really that's a pretty poorly written question IMO.

The question does not make much sense and I think you've gone about as far as anyone can go.

A) at least one character changes

Not necessarily! In any tree, of interest, it will be true that something will have changed but in theory, the only thing that separate two nodes is time. It is not impossible that your teacher was hoping for A to be the correct answer but it is not IMO. It is also very much possible that the only change in between two nodes relate to synonymous mutations (and other mutations that has no effect on the phenotype), hence not causing a change in any character fo interest.

B) all organisms die

Well, most likely enough time separate the two nodes that all organisms that were represented in the ancestral node are dead but of course they have reproduced, so that we actually have another node that descend from it.

C) Nothing can happen between two nodes

As you said, they evolve. So, things happen!

D) Monophyly, Paraphyly, Polyphyly

As you said, those terms make no sense here!

E) None of the above

If there is any intuitive answer to the question what has happened between two nodes is time has passed. I think the answer your teacher was expecting here is E. But really that's a pretty poorly written question IMO.