Since it's only 17" And I"m thinking about just making it a classic games arcade, I should probably just go vertical right? Is it dangerous or harmful in anyway (to myself or the monitor/arcade machine) to have a computer monitor in there all weird like and sideways?
Not generally.
You may run into the odd monitor that is completely unvented on the sides, and may have some heat issues if you leave it in the case though.
An even worse thing (that we do all the time anyway) is to lean the monitor so far back that its neck is straight down-ish, collecting all the junk that drops off the front of the monitor from use.
That, like vertical orientation, will "shorten" the overall life according to most manufacturers--but BOTH of those were done in alot of the original arcade cabinets, which use basically the same tube.
If I do go vertical should I just go with a 4-way joystick only?
I'd go with 1-stick, and have two buttons on each side of it.
That will let you play pretty much any classic vertical game, and will let people use whichever hand they are most comfortable with on the stick.
No buttons at all will limit your game selection drastically; and only one button will limit you somewhat.
There aren't many vertical games that use more than two buttons.