2004 Nissan 350Z Start up

That's called "idle flare-up" and is what all electronically fuel injected engines are supposed to do.

That's a completely different symptom. There's two common things to consider. One is the idle speed motor is sticking or the air passage is plugged, or the battery was recently disconnected or run dead, and the Engine Computer lost its memory, including what is commonly called "minimum throttle". Most cars have a procedure for the relearn to take place, and most of the time it just involves meeting a few specific conditions while driving. For example, with Chrysler products, you drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals. Until that relearn is done, the engine will idle too slowly and may be hard to start and may stall st stop signs unless you hold the accelerator pedal down 1/4".

Sorry for the delay. I saw your post yesterday, just in time for my laptop battery to die.

I only know the relearn procedure for Chrysler products. Some other brands are similar, and some don't show any symptoms. If you've been driving the car like this for weeks, you should have met the conditions necessary for any relearn to take place. The service advisers at the dealership should be able to tell you what, if anything, is required during the drive cycle. If they say there is nothing special you have to do, then you'll need to look at the automatic idle speed system. You'll need a scanner to allow you to command the Engine Computer to raise the idle speed to various settings to see if the system is working. Most independent repair shops will have aftermarket scanners capable of doing that. If the scanner says it's commanding a higher idle speed, but the engine doesn't respond, the idle speed motor is sticking or the air passage is plugged with carbon. We used to see a lot of carbon trouble, but that's much less common today with better additives in the fuel.

Common things to check are the rubber hoses that run from the steel fuel lines on the body to the engine, the o-rings around the injectors, the obvious steel lines under the car, and if your engine uses a fuel pressure regulator on the fuel rail on the engine, pull the vacuum hose off and check for fuel inside that hose.