What happens when ε-clos(X) contains duplicate elements? Example: (1+0)*1
Yes, your NFA is correct. You are probably making some mistake during your conversion to a DFA.

Note that after the conversion from the NFA to the DFA, the states will no longer be $S = \{A,B,C,D,E\}$. Instead they will be all possible subsets of $S$. So in your case $A$ is not giving two different transitions to $C$ and $E$, instead it is giving one single transition to the state $\{C,E\}$