hard start; rough idle. please help
 

I have a 2002 Mazda Protege. I bought a brand new battery a few months ago, so that isn't a problem. I also installed new spark plugs.
The car has trouble starting, especially when it is really cold out. I turn it over for 4-5 seconds before it finally, slowly, kicks in. When it is extremely cold out, it just doesn't start (remember I have a brand new battery). What do I have to do to fix this? I just want it to start normally.

The idle got rough at about the same time this started as well.

I really appreciate any help guys

Does it turn over at normal speed?
First thought is a poor ground - could cause low starter voltage, especially when it's cold out. And could account for a sensor reading if the ground is bad - which could cause the rough idle.

I know on some cars the crank angle sensor will go bad, which would prevent starting and cause rough idle depending on the failure. Though a number of sensors could cause the rough idle.

I assume if the check engine light was on you'd have mentioned it....?

Yes, it does turn over, the engine just doesn't kick in for about 5 seconds.
The check engine light is not on.

You think it may have something to do with the Idle Air Control Valve?
I really don't know.

If it's very repeatable (after x minutes of the engine being off) try this the next time... turn your ignigion on, so all the dash lights and all come on but wait 5 seconds before trying to start (crank position).
I've seen a similar issue to this if there's a fuel system leak, the injectors loose pressure and it takes a few seconds to build up enough pressure to fire properly... and the additional crank time also ejects the too rich mixture from the cylinders.
If an injector is leaking badly this could happen and cause the rough idle as well due to a rich mix.
Have you noticed any MPG difference since this started?