Entity set that has a relationship with itself
There's nothing in the ERD to constrain the model either way. Without suitable constraints, a child could be recorded as being born before his/her parent, a person could be their own child or parent, and cycles over any number of generations could exist.

For example, assuming the ERD is implemented as:

Person (personId PK, dateOfBirth, name, gender)
ChildOf (personId PK/FK, parentId FK)


we could record the following sets of rows:

Person (1, 2000-11-01, John, Male)   ChildOf (1, 2)
Person (2, 1970-05-23, Jane, Female) ChildOf (2, 4)
Person (3, 1985-01-11, Jack, Male)   ChildOf (3, 3)
Person (4, 1950-11-01, Joan, Female) ChildOf (4, 1)


There's nothing in the ERD to prevent any constraints either. As it stands, I would have to answer "yes" or "it depends on what unstated constraints you assume or implement" to both questions. If such constraints were required, I would recommend stating them explicitly.