The issue of missing work for illnesses of others in your family would fall under the Family Medical Leave Act (FMLA), and would require that you follow proper procedure for obtaining leave, assuming you otherwise qualify under that Act. For you to claim disability under the ADA, you would have to establish both a qualifying condition, notify your employer of that condition, and request reasaonable accommodation.

Given the circumstances of your case, you would benefit from consulting with an plaintiff-side employment lawyer in your state about (a) whether you have a claim based upon events to date, and (b) how to protect your rights.