The clue is in the volt dropping low even when driving, it's a duff alternator, if the diode pack(regulator/rectifier) is packing up it will cause all of this, I've seen one melt the solder points on the rectifier when it got too hot on the rollers, even more embarrasing it was a car I had a hand in!!
The regulator as you can imagine regulates the amount of current, the rectifier part converts the alternating current produced by the alternating current generator/alternator into direct current DC that the battery uses, as AC kills batteries, in doing so it uses a series of diodes, one way electrical valves, after time these leak and allow the battery to discharge through the alternator.
Annoyingly they will still always test perfect on a test for voltage or current, unless absolutely knackered, but rarely show up as bad.
Only way to see for def is leave alternator disconnected overnight.
tabetha