rough ride and cold start up issues

The front are struts and the rears are shocks. There are luckily plenty of options out there for you. If you have the original springs still not a bad time to replace those. Your choices are KYB, Tokiko or Bilstein depending on your budget and your desired results. All of these are better than stock options and probabbly in that order go for stiffness and handling. I have the KYB's that I put in a long time ago, they are not as stiff as others but do improve the ride. You can get a little lower stance with the lowering springs, also you can do coilovers but those are very stiff

For you cold issues there is a cylinder head sensor that I replaced when I had cold issues you should check that out first. If you have the original one it is toast.

My car used to run rough until it was at operating temperature. It turns out the air regulator was bad. Once I replaced that, the car now idles up to around 1500 to 1700 rpm and gradually decreases as it gets closer to operating temperature.