The red dot in your eyes is a sign of eye disease if you don't treat it well. It may be caused by the dryness of the eyes. You'd better use the eyes drops with anti-inflammation to get rid of it. Finding the clean cloth and hot water could also be useful. The warm compress is also regarded as the good way.
OMG, from your description I know your problem with your eyes is quite serious. But I still haven't got a rough idea of what it is for lack of detailed info. However, according to lots of cases around me, the red dot might just be a extravasated blood, which might be resulted from traumas. And other reasons are also possible, anyway, it would be wise for you to apply warm compress to your eyes regularly, and take medical actions against the problem.