First of all, congrats on the great deal on the rig!
Next, there's nothing dangerous about "all the amps" a 12V car battery can provide. 12V is too low a voltage to receive an electric shock -- it's impossible. If you've ever received a shock from your automobile, you must have come in contact with HV wiring when the engine was running, which is possible if you have leaky spark plug wiring/boots/insulators, etc. If the engine's shut "off," there's nothing to create a shock. The main reason automobile manufacturers settled on 12V was its excellent compromise between safety and efficiency (6V was also used, and is very safe, but less efficient). I cannot recall ever hearing of anyone receiving a real electric shock from 12V, the voltage is too low -- it doesn't matter that the battery can deliver 1000 amps, your body cannot even draw one milliamp at such a low potential.
There is real danger in working with automotive electrical systems in two regards:
1. Jewelry! People have received serious burns to their hands, arms, necks and so forth when metallic jewelry finds its way to short circuit +12V to the chassis of a motor vehicle. A ring, or bracelet (or whatever) worn and creating this short circuit becomes very hot, very fast and can cause serious burns. Obviously, wearing jewelry when working on anything electrical is not a great idea.
2. Explosion! Lead-acid wet cell batteries outgas hydrogen, an explosive gas. You don't want to be making sparks right near the battery, just in case. That's why advice about "jump starting" cars always recommend making only the "+" connection at the battery, and making the "-" connection elsewhere, like to the engine block, some distance from the battery, where any spark created is very unlikely to ignite hydrogen gas that may be wafting around the battery itself.
That being said, cars are safely jump-started by the thousands, every day, all over the world, and we very rarely hear about any accidents.
Mounting the rig vertically, on the passenger side of the center console, is a great idea, and one I use, myself. My own 2m mobile rig, a Yaesu FT-3000M, is mounted there and really does not intrude into the leg space of the passenger -- although occasionally a passenger will bump his or her leg against the rig. Oh well. It may be the only space you have.
If you're skittish about making the electrical connections, I'd recommend driving the car, with the rig pre-installed mechanically, into your local car alarm or car stereo installer and just having them make the connections to the battery for you. They do this all day long, they're really good at it, and cannot possibly charge much. The $200 is a bit steep! I've seen our "local" (here in Los Angeles) stereo installers charge $25 to do a ham rig installation, very professionally, in about ten minutes.
73 & good luck with the new rig!