A monitor in essence is just a device used to see what's going on! So yes, you can use that monitor for a PS3 as long as you have the right connectors (i.e. HDMI to HDMI, or HDMI to DVI and speakers, or HDMI to S-Video and speakers, or whatever).
In answer to whether you will get a better picture with an HDMI DVD player then the simple answer is no (but read on...). HDMI is just a connection type, it's a way of sending a signal. If that signal that is being sent is only SD (standard digital) then HDMI will carry a SD signal and the monitor will then display it. So if it's just a SD DVD player, but with an HDMI connection then there won't be any difference. However, there are some DVD players that have the HDMI connection and also have something called HD up-scaling. What this does is take SD and convert it to 1080i, which in effect means that you would have a better DVD player over your DVD drive in the computer.
Another thing about HDMI (high definition multimedia interface) is that it carries both video and audio, whereas S-Video, etc. only carry video and you'd need something else to carry audio. Clever HDMI.
*EDIT* By the way, using HDMI over SCART might give you a bit of a better picture compared to say SCART because it's a digital connection over SCART which is analogue, but it might not be worth spending the money on a new DVD player.
I hope that helps. If not, ask more