You can buy diodes online and sometimes at electronic supply stores. The diode costs can range anywhere from tens to thousands of dollars, depending on the laser you choose.
The diode will look like a small round light. It is likely encased in metal and will be positioned so that it can shine on the bottom of the DVD/Blu-Ray tray.
The driver should not be connected to any power source until soldering is complete.
The driver will have leads for a power supply. These leads will need to be connected to a battery pack or directly to the battery. If the leads need soldered, you will have to use a battery pack. You cannot solder onto a battery.
Glass lenses are better than plastic lenses, but they are more expensive.
It is a crime to point the laser at aircraft and/or motor vehicles.