Classic symptoms of leaky injectors. You'll need to replace them -- about $30 apiece via auto haus az dot com. Be sure to get new plastic holders and injector seals, too. Cheap parts, all, but get the seals and holders FACTORY ONLY parts - DON'T USE AFTERMARKET FOR THESE - and order them via parts dot com.

The reason for the 5 seconds rough running is that you have one or more injectors that are leaking gas into the cylinder after the car is shut off. The rough running is the engine working through the spilled fuel, and then settling down into a normal idle pattern once the excess fuel is burned off.

It's about a 2-hour DIY job to replace the injectors. DON'T try to "clean" them with Techron or other injector cleaner. They are worn on the inside, so no amount of cleaning will cure them. Just do the right thing and replace them.

Cheers,
Gerry