1. Most common size montior for a cocktail is 19", but you do sometimes see smaller depending on the setup.
That is incorrect. The most common size of MIDWAY cocktails is 19", most cocktails are actually 13". Go to the auction and look at the cocktail row sometime, pretty much everything other than the Midway ones will be 13". Also, cocktails with metal legs are generally referred to as "school desk" cocktails, and they are 13" ones, they aren't even deep enough for a 19" monitor.
2. Yes, you can, but why would you want to? The great thing about a cocktail cabinet is you can sit on any side you lioke, there fore, creating cointrols for both a vertical and horizontal side....
Most school desk type cocktails can actually take their 13" monitor in either direction, with the exception being the Atari ones that actually had the monitor offset to one side to make room for the larger boardsets.
Many people might not want 4 control panels, because a cocktail with 4 control panels takes up almost an entire room with the 4 chairs, either that, or you are moving it all the time.
3. The vertical games would have black bands on each side, and truly just wouldn't look right.
This is true, it isn't too bad on a 19" monitor, but a vertical game on a 13" horizontal monitor is tiny. I suggest picking a monitor orientation and control type, and sticking with it.
If you are going the cocktail route, then vertical is really the prefered way to go, there are simply not a lot of horizontal games that have cocktail mode, but most vertical games do.