Agar is often used in place of gelatin (which is derived from collagen). In the UK its possible to buy vegetarian Jelly (jello, I believe its called in the US) which is agar based. Agar comes from seaweed.
Probably not because I know my highschool didn't provide anything other than salads and PB and J. They didn't actually have vegetarian/vegan entrees. We pretty much had to fend for ourselves in that respect.
Colleges may have to though. My school has a vegetarian entree for every meal sometimes it's vegan but sometimes not. They also have vegetarian meats and cheeses for the sandwich bar.
Either way, if you are concerned about eating vegetarian at school, your best bet is to probably pack your lunch.
Vegetarians do not eat fish. Vegetarians do not eat any animals.
Water chestnuts come from a plant. Therefore, they are suitable for vegetarians.