What is the make and model of the laptop? That's important to know so we can see exactly what hardware you're working with.
Regarding issue #1:
Do you have a screen saver active? If so, does it ever come on?
What programs are running on your computer at any given time? Some programs won't allow the computer to go into automatic standby or hibernation and often won't allow a screen saver to come on. For example, some media players, P2P programs, or other programs that actively use the hard disk.
Issue #2:
This might also be a result things running in the background. Depending on your power settings, there are different steps to powering down automatically when not in use. The following is an example and does not apply to all laptops and/or settings:
First, screen saver. Second, screen off and hard drive off. Third, system standby or hibernation.
The third step usually requires the power button to be pushed to turn it back on, but some computers support pushing keys or moving the mouse. Consult the laptop's documentation on that. Have you tried just pushing it once and not holding down the power button?