Example of impossibility in the nested hierarchy?
Your view of evolution is wrong


  If an organism had characteristics that came laterally (?) across the nested hierarchy, as opposed to up or down through its ancestry, that would falsify the hierarchy


The expression "falsfying the hierarchy" does not mean much, so I won't comment on it.

Note two things

1) Convergent evolution
 - It is not so uncommon that different lineages evolve similar traits when facing similar selection pressures. This is called convergent evolution.

2) Horizontal transfer
 - It is not so uncommon that pieces of DNA find itself (via a number of different mechanism) in a different lineage. A gene can therefore jump from one lineage to another one. This causes the gene tree of a given gene to not match the species tree. Other processes can cause such mismatch though. This is called horizontal gene transfer. 

So, in short, no your view of what you call "impossibility in the nested hierarchy" is wrong.

Intelligent design

Intelligent design is not a scientific hypothesis. It means that it does not make logical sense and it does not make clear predictions that can be tested. By contrast, the theory of evolution is a set of hypothesis all of which that have been formally tested have shown to be confirmed by evidence (that's why the theory of evolution is called a theory; 
see here for the semantic). 

It is therefore impossible to say whether intelligent design is correct or wrong because it does not say anything. One cannot compare the predictions of the theory of evolution with the ones of intelligent design as intelligent design does not make clear predictions. One can only argue why the theory of evolution
is a set of (confirmed) hypotheses while intelligent design is not a hypothesis (or set of hypotheses).