Bloodshot eyes are an inflammation or rupturing of the capillary vessels contained in the covering or outside layer of the eye (called the conjunctiva). That inflammation can be caused by all sorts of problems and can even be caused by too much time in the sun. The results can vary, depending on situational severity, from slightly accentuated vessels to blood-red "demon eyes." This means, of course, that the treatments may vary as well.
The term "lazy eye" isn't very specific and therefore can't be talked about as one problem with one solution. Strabismus is the condition most people somewhat incorrectly associate with the term. It can manifest as exotropia and esotropia (goggle-eyes and cross-eyes), in which one eye deviates outward, inward, or both. The term "lazy eye" is more correctly associated with how the brain makes one eye dominant over the other (amblyopia). It can be caused by strabismus, uncorrected vision, or cataracts. These kinds of conditions are usually discovered in early childhood and can be corrected on their own. If they do not, I have laid out a few of the most common remedies that you should discuss with your physician or ophthalmologist.
Pink eye (also known as conjunctivitis) is an infection or inflammation of the conjunctiva – the membrane that lines the eyelid and eyeball. This problem isn't cause for alarm and it will usually clear up on its own in a week or two; however, pink eye symptoms are very annoying and disgusting. You can expect redness, itching, burning, sensitivity, swelling, a gritty feeling under the eyelids, and discharge from the eyes.