SWR would probably improve a bit if it was centered. Since these are base loaded 1/4 wave antennas, the ground plane is necessary for their operation.
Proper grounding is necessary for the antenna to work right, but for dealing with noise, grounding at the antenna isn't the only place. You should ground the radio case, also, and not rely on the negative power lead.
As for noise, a properly installed antenna, including a ground plane, will work better, and the amount of stuff, including noise, you hear will increase. On the other hand, if you are talking about ignition noise, well, grounding everything correctly will help that to.
As for the cigarette lighter for power, the proper way to power a radio is from a dedicated circuit direct from the battery. Fuse the leads as close to the battery as you can. Run the power leads carefully, using split loom tubing and grommets to protect it from damage. You can power it off the cigarette lighter, but it's a really good way to induce noise into your radio. The circuit that feeds the cig. lighter isn't really designed to provide good clean power. The circuit may be shared with other accessories, run near noisy components (blower motor for heater), etc.
Your best performance will be to run it directly from the battery. I've done this with all my installs and I don't have any noise or alternator whine issues.