Penrose Conformal diagram for flat 2-dim Lorentz space-time

Question

I have the following metric

$$ds^2 ~=~ Tdv^2 + 2dTdv,$$

defined for

$$(v,T)~\in~ S^1\times \mathbb{R},$$

e.g. $v$ is periodic.

This is the according Penrose diagram:

Question 1) Is the Penrose diagram that I have drawn correct?

Question 2) How do I see (using the Penrose diagram) that two different spacetimes have a similar global and causal structure?

Question 3) What kind of global and causal structures does a Penrose diagram reveal?