Of course you're allowed to press thrice weekly. I've had great results benching the same way, using a heavy/medium/light protocol with superb results. The press would likely give you great results, too.
Yep. I did much the same thing for about a year - didn't have a bench and didn't want to buy one. It can be done, but I can tell you honestly that your press won't progress as well as it would if you were incorporating bench presses or adding in some LTE work. Can you do dips?
Technically I can, but it's an unstable surface so I prefer not to.