Regarding Caesar murder. Suppose we have a coordinate system at the center of the galaxy, for someone living near Sag A*. They're at rest w.r.t to Earth, this is Frame $S$. So, right now, their point on their world-line is:
$$ E_1 = (t=0,x=0)_S $$
where $t$ is measured in years, and $x$ in light years. The $x$-coordinate points to Earth. Caesar's demise occurred at:
$$ E_0 = (-2065, +26700)_S $$
It's clear that the murder was in their past (as we share the same clocks):
$$ \Delta t = t_1 - t_0 = 2065{\rm y} $$
although the event remains space-like separated:
$$ \Delta s^2 = \Delta t^2 - \Delta x^2 =
(-2065)^2 - (26700)^2 \approx -(26630)^2 $$
They will not be able to know these Earthly events for another 24-ish ky, nevertheless,
if a message could be sent instantly to Earth, Caesar's fate remains seal.
But, now they hop on a spaceship and accelerate to a mere 8% the speed of the light moving away from Earth. Their coordinate in their new frame $S'$ is:
$$ E_1 = (0,0)_{S'} $$
of course, but now compute Caesar's murder:
$$ E_0 = \big(\gamma(t_0+\beta x_0), \gamma(x_0 + \beta t)\big)_{S'}$$
$$ E_0 \approx (71, 29571)_{S'} $$
Even at mild 24,000 km/s, Caesar's murder is 71 years in the future. If they can send a message to Earth at any speed greater that $29571/71=425c$, Caesar can be warned.
Note that I used $\Delta x'$ to compute the speed, not the standard distance to Sag A* of 26700 light years. There's no preferred frame, so I can use whichever one I want. Once a message moves faster than light, even by a tiny fraction, there is a frame in which it is faster still. There is a frame in which it is instantaneous, and there are frames in which it goes backwards in time.
So: no FTL communication allowed.