Well, as a matter of fact, taking exercises would not lead to black eyes. Generally speaking, black eyes are resulted from some bad living habits including staying up or playing too much video game without proper rest. So, I can assure you that it would not cause black eyes. Maybe you should check if there is anything that are bad and try to quit some bad habits.
Nope, you should not do that. A black eye means there is injury to the face or the head because the blood and other fluids are collected around the eye area to appears it darkness. However, the eye itself is not injured, just there is a bruise around the eye. When you doing exercise, especially strenuous exercise, it would easy to make further injury on head or face area. It would make your condition worsen and possible to cause unexpected result. Hence, you should avoid injurious activities until it has healed.
Well, generally speaking, exercises can not lead to black eyes, unless you have got injury because of exercises. So you should look for some other reasons. And in most cases, black eye is caused by the condition that blood and other fluids collect in the space around the eye. From some experts, it is often caused by the injury to the face or the head. So it may be like your situation. For your situation, I will suggest you stop having exercises and see an eye doctor. But on the other hand, you also can do something by yourself. For example, you can try using ice to reduce the swelling caused by a black eye. Also, warm compress will also be effective for your black eyes, for they can make your blood flow smoothly. So you can just put a warm cloth on your eyes. Anyway, you can have a try. Anyway, having a good rest is also essential.