Let me complete Amir Sagiv's answer by answering parts 1 and 2, and clarifying a bit the comment of Tim Carsonof Tim Carson.
Let's tackle (2) first: assuming your local wellposedness result, it is impossible to have $u\in C([0,T^*], X)$ but $u\not\in C([0,T^*+\epsilon],X)$ for any $\epsilon > 0$: this is because you can solve the initial value problem starting from $t = T^*$ with $\tilde{u}_0 = u(T^*)\in X$ and, applying the local wellposedness result, arrive at the existence of some $T' > 0$ such that $\tilde{u} \in C([T^*,T^* + T'],X)$ that solves the equation.
Similarly, to answer (1), you see that your local wellposedness result actually implies that if $u(T) \in X$ you can extend to a solution on $C([0,T+\epsilon),X)$.
Morally speaking: that you have a local well-posedess result states that any time in the domain of existence has to be an interior point, and hence that the domain of existence should be open. The main difficulty in most global-existence-type results is in demonstrating that a solution that is known to exist on open intervals of the form $[0,T^*)$ can be extended to intervals of the form $[0,T^*]$.
