Dont ever go back to that tyre shop again.
165's on one end of the car, and 185s on the other could create some intresting handling if you push it hard enough. Basicly you've just changed the whole balance of the car beyond its design limits. However since its a Neon, I wouldn't worry to much, as you dont sound like much of a racer, and Neons arent reknown for thier great speeds.
Regarding the bumby ride- check the tyre pressure, a lot of tyre shops over inflate tyres. With such a small size something around 26-28psi should be alright on the 165s, and maybe 28-30psi on the 185s. You can go lower if the ride really bothers you, but it will hurt tyre wear and millage if you goto low.