You pretty much nailed it, except for the car "looking right".

Here's my advice (and yes, you did ask for it)

1. Leave the stock suspension and use oem wheels and tires.
2. Keep the 18s and lower the car to reduce the unsightly wheel gap.
3. Go back to oem wheels and tires and lower the car to get rid of the still somewhat unsightly wheel gap with stock running gear.

Thats it, you can't go throwing huge sidewall tires on 18s and expect that to "fix" your problem. The largest you could really go would be something like 225/40/18 or 215/45/18 (the latter of which would likely be borderline acceptable).

And get rid of the neons.