1. PRAM battery will cause the date and time settings to be lost when the mini is powered down
2. The crashing can be caused by a corrupt OS.
3. Slow boots and shutdowns can be caused by a faulty hard drive and corrupt OS

What to do?

The battery can be easily replaced but the mini is not the easiest thing in the world to get into. It requires the use of a Putty knife (Scraper). If you feel up to the task there are a number of places around the internet that will show you how.

The corrupt OS can be replaced by using the OS X installation disk. Booting from it and choosing the Archive and Install option. This will keep your data intact and place a new copy of the OS onto the drive.

Before you go there, boot from the installation disk and run the Disk Utility that you will find on there.
Choose the Repair Disk option and then run Repair Disk Permissions.

Reboot normally and see if you still have the problems. If yes, Archive and Install.

Good Luck, let us know how you get on