Can a bad battery cause a rough idle? NO, as long as the alternator is working.
As soon as the engine starts running the bulk of the loads are handled by the alternator. It's only when the alternator can't make the required amperage that the battery is called upon.
Get a digital voltmeter and measure across the battery terminals before starting (~12.7V) and then start the engine measure again. It should be between 13.5 AND 14.5 VDC.
Now switch the voltmeter to read AC voltage. Check across the terminals again. The voltage you may measure is called ripple and if it's more than, say, 0.2VAC you have a diode either going out or dead.
i had something very similar happen to me
if there is enough juice in the batt to start it up
that is your highest demand out of it so it is most likely ok.
heat and cold will effect the diodes when they are weak
try changing out the alt with a known good one
