I will try to answer question 1. on my own as good as I can:
Consider two cases:
1.) $\hat{T} \cap E(\mathbb{Z}) \neq \emptyset$Let $Q$ be an element in this intersection. Then $-Q \in \hat{T}$ and adding this to $Q \in E(\mathbb{Z})$ we get by hypothesis on $\hat{T}$ a point in $E(\mathbb{Z})$, so
$$ O = (-Q) + Q \in E(\mathbb{Z})$$
which is a contradiction, to SAGEMATHs definition of integral points $E(\mathbb{Z})$.So this case can not happen.
2.) $\hat{T} \cap E(\mathbb{Z}) = \emptyset$
Then by Nagel-Lutz Theorem, either $\hat{T}=1$ or each $Q \in \hat{T}$ is of the form $Q=(\frac{m}{4},\frac{n}{8},1)$ and has $\operatorname{ord}(Q)=2$. This means in the latter case that $\hat{T} = C_2 \times \ldots \times C_2$ which leaves by a theorem of Mazur only the cases $C_2,C_2 \times C_2$ open.
All in all, we get three possible cases:
$$\hat{T} = 1, C_2, C_2 \times C_2$$
of which I have seen examples for the first two. Is there an example for $\hat{T} = C_2 \times C_2$?
