Rough idle

96 Seville, northstar. I park in garage. Sometimes I start car, back out into driveway and use garage for work area. Next time I start car to move it back or go on errand, the motor runs rough for 10 sec or so. Car did have headgaskets replaced 6 yrs ago. Car runs fine normally, uses no coolant. Does this sound like an excess fuel issue? Perhaps a leaky injector issue? Fpr has been replaced. Or do some cars just not like a 10 sec start/stop sequence? I have noticed this issue for years. It’s is not a new issue.

“Or do some cars just not like a 10 sec start/stop sequence?”

I think that’s the answer.
If the engine behaves properly after a “normal” drive of a couple minutes or more I don’t see a problem here.

I did drive into garage and shut off a few times. Than next time I drove, it really had a crappy initial idle. So that was consecutive 10 sec runs. Don’t do that anymore. I just know now that a 10 sec move wil lead to a rough next start.

Here’s what I suspect is happening:
When you shut off the engine one or more cylinders is left with unburned fuel/air because it was on its intake stroke.
If the engine is cold the unburned fuel condenses on the surfaces, including the spark plug, and that causes a momentary misfire.
In a cold engine the fuel/air mix is slightly rich, which adds to the problem.

My 73 Ford truck does this too. Runs a little rough for 10-15 seconds after being shut off from a cold start. I suspect the choke remains closed and the carb pump is shooting more gas than needed and the result is too much gas in the engine on the next start. I just ignore it, as it runs smooth again in short order.

In your case, either @circuitsmith 's theory is the reason, or another theory, perhaps the engine coolant sensor is not accurately representing the overall engine temperature as very little coolant has circulated during this operation. This could confuse the ECM to inject more or less gas than is needed. Once the coolant circulation stabilizes, the ECM smarts up, and figures out how much gas to inject.

If you remain concerned with the head gasket, mechanics have an assortment of tests to say “yes” or “no” to that. They might start with a intake manifold vacuum test and a cylinder compression test and depending on the results, go from there.

This could be caused by a loss of residual fuel pressure where the fuel system is not maintaining pressure while the engine is inoperative.
This results in rough running while the air is burped out of the fuel rail and lines.

In this case, a 96 with a 10 second rough idle-don’t fix what isn’t broke.