Minimized Apps: Do they consume more battery? If I keep one or more app minimized in my Android Phone (Samsung Galaxy Y) will my mobile's battery will be consumed faster than if I had not minimized any application at all?
If an app does not run in the foreground, it is usually "frozen"; this means, its memory structure is kept (so you can "recall" it faster, and no additional I/O + CPU is needed to restart it) -- but gets (almost) no CPU-cycles assigned. There might be some exceptions when it needs to do things while backgrounded, but usually that would require the app to use a "service".

Thus, the "overhead" is minimal -- compared to the need of restarting it every time the user needs to access it. As for memory, the Android system is quite efficient in maintaining it: To explain it in easy terms, apps are put into different groups having different "priorities". Together with some other criteria, this decides which app can be killed if the system runs "Out Of Memory": The OOM-Killer (again, "Out Of Memory Killer") comes into action then and checks: Obviously, essential system processes should not be killed at all, the users foreground-session (which he is working in) should be the last to consider. But "closed" apps are first candidates, "background apps" the next, and not-so-essential services come after those.

Again, that is a simplified way to put it -- in reality, this looks a bit more complicated :)

BUT: if you think of task-killers to "improve battery life": Forget about that before starting to think about. They are usually rather contra-productive, as they work against the system. Few exceptions apply in terms of a malfunctioning app (e.g. consuming all your CPU and there's no other way to stop it).