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.

If turning the idle mixture screws from 1.5 turns to 2.5 turns out from lightly seated does not have an effect on the idle quality, then the engine is not running on the idle circuit or you have an air leak that is causing the power circuit to open, something like that. You could probably read a writeup or two about setting lean best idle.

Good Luck!

R.