No problem, and me too as I had left for the night when you returned.
There are only two real considerations here and that would be an unmetered air leak (vacuum leak etc) which will cause the idle to go high, or a IAC (idle control valve) failure.
Since you are seeing (if I understood correctly) high idle and lurching etc when cold, but once it is warmed up it gets better, this would most likely not be vacuum related since that would not change when warmed up, but would likely be IAC related as this is incredible temperature dependent.
Specifically, the IACs purpose is to adjust the idle to compensate as needed for different temperatures of the engine. Having it idle high when cold them more normal when warmed up just indicates the IAC is likely stuck opened too far, and once the engine warms up and can use more air it is now in a better operating position where it is stuck, so drivability improves.
There is a relatively easy test you can do on this as well. With the engine idling (preferably warmer so it is lower), turn the AC on and watch the engine speed. Under normal operation when you turn the AC on the computer should idle-up the engine about 100-200 RPM, then when the compressor engages the load will drop the speed down again.
When an IAC is not working the dead giveaway is that when you turn the AC on the idle-up does not happen, and then when the compressor kicks in the RPM goes down as there is no compensation for the load occurring.
Having an improvement when you had the dead battery makes sense for this too... when the battery is depleted (or disconnected), the computer loses its idle adaptation logic and readapts to a zero point condition (and normally the IAC follows suit if it is working). If the adaptation was considerable and it was lost, it is going to assume there is more air flowing in that normal (since it is expecting a "clean" throttle body rather than obstructed with gunk it adapted for). In your instance since the idle is high and there is too much air going in, this looks to the computer like you just cleaned the throttle and cleared adaptations and it would run better temporarily.
I would be doing that IAC test and see how it responds, and ultimately I would probably end up replacing the IAC (mounted on the throttle body) and giving the throttle a good cleaning inside while I am at it.