The regulator is built into the alternator. Many alternators are replaced every day that could be properly repaired by replacing the alternator alone. You don't need to worry about this one as long as the battery is charging properly.
I have only had the alarm go off on my 300E once or twice. I even forgot that there was an alarm on the car. I think I locked it with the passenger door still open and it took my wife so long to get her purse, get out and shut the door, that it went off. I expect that because the battery had probably been disconnected, the alarm had to reset itself somehow. You may have no more problem with it and you will forget the car has an alarm.
'84 Euro 240D, 523K miles
'88 300E 5 Speed
'81 300D Daughter's Car
Over 800,000 miles in