Are there a lot of quasars in the universe?
There are about 12,000 known quasars today. I'm sure that as our telescopes get better, that number will go up. As a guess, I would estimate a lot.
The steady state theory was disproved by observations in the sense that at larger distances, the Universe doesn't look the same as nearby. This means the Universe has changed over time, directly contradicting the main assumption of the steady state theory. - I am not sure to what extent quasars were involved, but quasars do tend to show up more frequently at greater distances, i.e., in the early Universe.