Applying a 1-qubit gate to entangled qubits
I think you're confusing people talking about what must be done theoretically (to compute operations on qubits on paper) with people talking about what must be done experimentally (to operate on qubits in the real world).


  Do the answers contradict themself?


No. The person who appeared to be saying "you can't apply the operation" just meant that the algebra you do to apply the operation won't start by factoring the state. When doing algebra you can't apply the operation in the simple way that you'd use when the qubits factor.


  Why can't we just do $A|0\rangle \otimes B |0\rangle + A|1\rangle \otimes B |1\rangle$?


You can! $A|0\rangle \otimes B |0\rangle + A|1\rangle \otimes B |1\rangle$ is equivalent to $(A \otimes B) (|00\rangle + |11\rangle)$. To apply a Hadamard operation to one qubit while leaving the other alone, set $A=H$ and $B=I$.


  Alice has only 1 qubit, but U⊗I acts on 2 qubits. How does it work physically?


Alice just does whatever she would do to apply $U$ to her single qubit. That applies the operation $U \otimes I$ to the system of two qubits.