Question

Let $\mathbf{G} = \langle G,\cdot,\mathcal{T}\;\rangle$ be a topological group. Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$. Assume ${\mathbf{e}}$ is closed in $\langle G,\cal{T}\;\rangle$. Then, I have managed to convince myself that:

1. ZF proves that $\langle G,\cal{T}\;\rangle$ is regular Haudorff.
2. ZF + (Dependent Choice) proves that $\langle G,\cal{T}\;\rangle$ is completely regular.

My questions are:

1. Are those right?
2. Does ZF prove that $\langle G,\cal{T}\;\rangle$ is completely regular?
3. If no to question 2, does assuming one or more of following suffice for ZF to conclude that $\langle G,\cal{T}\,\rangle$ is completely regular?
   1. $\mathbf{G}$ is two-sided complete
   2. $\langle G,\cdot \rangle$ is abelian
   3. Countable Choice