If person living with me doesn't wash her hands after using toilet (for peeing, i'm sorry i said it) could she transmit some serious diseases to me?

Unlikely: Urine itself is usually sterile, and infections associated with STD's generally do not survive a dry environment and require direct contact to cause infection. On the other hand, respiratory infections and gastroenteritis viral infections are very contagious and would most likely be spread with house hold contact, especially if hand washing is not done after coughing/sneezing and toileting.