I can give partial answers to 3-4. 

One definition to finite-time blowup is: there's a $T_c>0$ so that $\lim\limits_{t\to T_c ^-} \|u(t,\cdot)\| = \infty$. What is the appropriate norm? When $F(u) = |u|^{2\sigma}u$, for example, there's both an $L^{\infty}$ and $H^1$ blowup for certain values of $\sigma$ and initial conditions in the respective normed space. For more details, see "The nonlinear Schrodinger equation: Self focusing, singular solutions and optical collapse" by Gadi Fibich, specifically chapters 5.5-5.6.

From here it is apparent that there might be local well posedness in the aforementioned norms and a finite time blowup for the same initial conditions. So, it seems to me that the answer to question (4) as I understand it is that you can't generally use local well posedness to prove the global one.