Pushups are mainly pectorals and triceps. You have to have a much broader regiment if you want to increase your weight. Eat a lot of pasta, breads and potatoes, along with plenty of meat for protein. Protein and carbohydrates are what put on mass the fastest.
Where do you live? In the states, that is not an issue, at all. I eat a lot of red meat. I'm in upper Michigan now and shot a big deer. I'll be eating that for another two months, then it's back to beef.
Yeah, I've heard of that. I personally don't know what else is available to you that's safe, but I would start looking at alternatives. There's not a lot of protein in chicken. Lamb maybe?
Lamb is delicious. Very tender. It has a different taste than beef, but then it's a different animal, don't be grossed out by it cuz it tastes different, enjoy the experience. If you've never had it before, then I suggest trying lamb chops for your first try. Very fresh and clean tasting. A lot of people eat it with a mint sauce. Experiment.
No situps and crunchs will do the best, but as for being manhandled as you say. Being small works for your advantage. If you have to push the sides of their hips, they will go back as long as you use enough force, since that's their center of gravity. For tall people their knees are another spot to target, such as a solid strike to the side of the leg. No matter how big or small you are a strike to the solorplex will put them back, along with a genitalia hit. I personally am 6'2" 225lbs and have had to work on my knees for years for problems with them. I'm not going to condone fighting but I'm not going to condone bullying either. You should bring this up to teachers or management pending were your at.
A situp is laying on your back, knees bent in the air feet under a solid surface, and touching your elbows to your knees, and a crunch is the same thing just less stressful on the back. YouTube will be able to show ya.
Pushups, dips, and pullups are my main workouts. Ive been doing them for years now and i haven't really gained tons of muscle i just have A LOT more definition in my arms, back, and chest. im 5'8 and have stayed 165 for about 4 years now
also look up calisthenics on youtube or google. free running is in my opinion the best form of exercise