On the application that I am filling out, they have split up the name section into three parts much like any English application, but instead of 'Last name' it says 'Family name'. I usually put my last name and add a 'jr.' at the end, but my last name is my family name, not 'jr.' So is it ok for me to put my last name and add a 'jr.' for the family name field, or would I put the 'jr.' elsewhere.
Your mother (and a possible sibling) is a Watson, and your last name is the same as hers. I doubt she is a "junior" so I would not put it as last name.
You and your father are both Richard, but you are "Richard Jr." so I would add it to your first name.
Most people in the USA know a lot of "Richard Jr"s or some people who just use "Junior" as their first name by default.