Re: What proof would be necessary to prove steady state theory

In the 1930s and 40s most cosmological discussions focussed on
fairly philosophical issues, in the absence of very much data.
Through the 1950s and 60s, two particularly important developments
occured which swung the debate decisively in favour of the Big Bang, 
and against the Steady State.

First was the comparison of observed abundances of different chemical
elements with the predictions of the two theories.  In particular, in
the Steady State theory all matter was presumed to start out as hydrogen
(ie. single protons and electrons) and be cooked into heavier elements
in stars.  This conflicted with the observation that 25% of the observed
matter in the universe was found to be helium compared to about 75%
of hydrogen (by mass), plus a smattering of heavier elements. The 
Steady State calculations suggested that only 1 or 2% of the helium
could have been produced in stars without greatly over-producing other
elements.  By contrast the Big Bang calculations included the 
synthesis of helium from hydrogen which should take place in the first
few minutes of the universe, and found very good agreement with the 25%
helium observed.

Secondly, and perhaps most importantly, the Big Bang theory made
a prediction that the Steady State theory did not, that the universe
should be filled with low energy radiation which was a relic of
the primordial fireball.  In 1964 this radiation, now known as the
Cosmic Microwave Background Radiation, was discovered
with essentially the properties which had been predicted.  Since the
Steady State theory made no similar prediction, the credence of
the Big Bang received a big boost!

Since that time, observations of the chemical elements and the microwave
background have improved greatly, and overwhelmingly favour some
kind of Big Bang (although there are a number of different theories
which broadly fall under the Big Bang umbrella, which differ in detail).
We have also seen other evidence for evolution in the universe, for
example distant galaxies (which we are seeing as they would have
been far back in time) appear quite different from their nearby
counterparts, which also speaks against the Steady State.

Regarding your second point about rotating galaxies, there is evidence
that the rotation of the Sun around our own galaxy has an influence
on mass extinctions, in as most of the larger mass extinctions in
the past 600 million years have coincided with times that the Sun (and
Earth) was passing through one of the Milky Way's spiral arms.  This
is not too surprising since in the sprial arms the Sun is more likely
to pass close to other stars, disturbing the cometary (Oort) cloud
around the Solar System and sending many new comets in toward the Earth.
In the spiral arms the Earth is also more likely to encounter nearby
supernovae with similar catastrophic effect.