First off, you cannot catch stress from other people, but you can get stressed by them. In other words, if they are down all the time, that can start to make you feel down too.
We all need some stress in our lives, but too much of it will make us feel ill.
When you say that you "don't seem to have mental issues, just physical issues", stress in fact is a "mental issue" and being under a lot of stress can manifest itself with physical problems, as well as feeling sick and having loss of appetite.
As the symptoms you describe can also be caused by other medical causes, it would be wise for you to make an appointment to see your doctor to make sure that you do not have any other underlying medical problem. The symptoms that you mention could be depression and your doctor will ask you questions to ascertain whether you have got depression or whether it is stress due to your school work load. He may also take a blood test to check whether you are anaemic or not.
In the meantime do a timetable chart by prioritising the things that have to be done urgently and first and those that can wait. Allow some time for "me" time where you can forget about your homework and studies. Go for a walk in the fresh air, meet some friends for an hour or so, do some of your favourite sport or go swimming. Physical activity is also important and will help you to clear your mind and allow you to concentrate better.
Make sure that you have a healthy and balanced diet with lots of fresh fruit and vegetables and drink plenty of fluids so that you do not get dehydrated. If you are OK with milk, drink milk which is good for your growing bones and teeth, also drink plenty of water. Keep off or cut down on fizzy drinks and caffeinated drinks. Cut down on junk foods, sugary snacks and processed foods. If you need a snack have a healthy snack like some nuts, an apple, a carrot or a stick of celery or even a piece of seeded brown toast with some peanut butter (if you are not allergic to nuts).
Make sure that you go to bed at a reasonable time every day. Have a warm relaxing bath to help you to unwind and if it helps and you like warm milk, have a milky chocolate drink before going to bed to help you sleep.
Getting dehydrated can cause headaches, so do make sure you drink plenty of fluids - but not fizzy or sugary or alcoholic drinks.
You can purchase roll on lavender that you can roll on your forehead and your wrists that can help to calm you down.
I agree with jemma that stress itself is not contagious. Being around people who are stressed can make people feel stressed just like being around depressed people can make one feel depressed and being around happy people can make a person feel happier.
As well, I agree with jemma in regards to stress being a mental issue that can manifest itself in the body with very real physical symptoms. Insomnia, aches and pains, chronic headaches, back aches, restlessness, stomach/digestion tract issues and more all are side effects of stress.
I would also agree that you should talk to your physician. A proper diagnosis is essential and a regimen of eating right, proper hydration, and exercise may be the easiest way to get back on track. As well, evaluate who you are spending time with and figure out if they are the cause of some of your stress. (You said above you felt they were.) Spend less time with them and more working on you. Prioritize the things you must accomplish and also spend time doing an activity you enjoy doing.