Did you catch the "How to install a Mac Mini" FAQ in the FAQ Emporium? That will answer all of your Mac Mini specific questions.
Most of the things you are asking are not Mac specific. This is the good news. The bad news is that you don't have experience with car audio or electrical wiring. However, you can find out an awful lot by reading the FAQ's, such as "How to install your Opus" (good advice, also fairly applicable to the Carnetix P1900), "How to find the ignition wire in your car", and so forth. There's also one in there about "How to get audio to your car speakers".
In addition, you can look at 0l33l's (that's zero-l-3-3-l) installation and talk to him. He's a Lexus freak and can probably point you to the right Lexus forums so you can figure out the aftermarket aspects of your install.
Also, post a pic of your dash here. You'll get lots of suggestions about where to put the screen.
Lastly, do you know anyone with car stereo installation skills? The screen will be the most difficult part, unless you go with a motorized foldout touchscreen.