It's better not to put it over the back of your chair because it can easily be stepped on or spilled on. By the way, your purse, glasses, shawl, camera etc are not allowed on the dining table. They go under your chair, at your feet or can be tucked in the small of your back while your seated. I can't afford to pay for a limo and a boat cruise and food. If you know him well enough, just ask if he would mind driving and if he's a gentleman, he probably won't expect you to pay for gas.
Proper etiquette dictates that who ever did the asking also does the paying.
However, everyone knows that graduation is very expensive and it is difficult for one person to handle all the costs. You of course will have to pay for your own tux and you should give her some small gift or a corsage. She is most likely expecting to pay for her own ticket so you could let her and pay for her share of the limo or something instead. Most girls would tell you that romance is about remembering the details and small thoughtful gestures- not how much you spend. Be a gentleman the whole night, tell her she looks great, hold the door, listen to her, hold her hand and pay attention to only her.