Yes, your eyes color can change with different emotions. Generally speaking, when you are clam, your eyes' color can change to grey. And they become Light grey and light blue, when you are sad, concerned, worried. According to the research, Color change is caused by blood pressure. If you embarrassed or excited, for example, your blood goes faster through the veins and if you're calm your blood then goes slower. This is just like how your face becomes red when you're embarrassed, your eyes change color. So it's very normal to have this situation. And you should not be too worried about it. In my opinion, controlling the emotion is a good way.
Yeah, it is possible. When you are sad, you may cry a lot which may to some degree make your eyes get red. The bad emotion will influence your nerves in the body which may indirectly decline your immune system. That is to say, your eyes may get infected easily because of the dust or sand. Then the color of eyes may change to warn to you that there exists the problems for your eyes.