Direct evidence for an accelerating Universe came from observations of type Ia supernovae by the High-Z Supernova Search Team (Riess et al. 1998) and by the Supernova Cosmology Project team (Perlmutter et al. 1999).
Their research showed that remote supernovae are 10% to 25% dimmer and therefore further away than expected as compared to the nearby (local) ...
the expansion of the universe is relative to where it is being observed from. Much like dots on a balloon. Its easy to point to the center of the balloon (like what the picture shows) because it is only a 3d object. But for higher dimensions we don't have the spacial awareness to understand where than center of the universe is (if there even is one)
There are theories which suggest that the universe is like a giant spring, constantly expanding and collapsing. Its possible that pre-big-bang, there was another universe which collapsed, only to re-expand into the universe we know now.
But in reality we have no idea
This is a difficult question to answer. Physics really starts after the big bang. Scientist don't know about the laws, if any, before the first instance after the big bang. The time before and during the big bang is really the province of philosophy.
That diagram does not depict the entire universe. At most, it depicts the history of what is now our observable universe (specifically, a 2D slice through it), with us at the center only because we're observing it. Someone at the furthest reaches of that portion of the universe would see us at the furthest reaches of their observable universe, and themselves ...