In particular, "Preview" and Finder's "Get info" windows always open on the far left of your desktop. This is incredibly annoying on multi-monitor setups if you are using Finder on the opposite side of your desktop.
I don't know why. The implication of your post is that you'd like to change this behavior. Unfortunately, for now, I think you're stuck with it. Make a suggestion to Apple. Maybe they'll address it in 10.6.5 or something.
Not all apps remember their window positions because the functionality isn't system-wide. Meaning that the developer needs to build that functionality into the app (it's simple to do) but in some cases the developer (Apple in this case) feels that it is not necessary.
In many cases, I have had luck with opening an app, moving the window where I want it, and immediately closing it. Seems if you do any "work" while the window is in its new location, it will forget that location and go back to where it was when you opened it that time.