Because they have eradicated the illusion of self
(identification to body and mind as 'self' or identification with a
separate experiencer), no, they do not feel fear. As my friend
Thusness said, the three important factors of self liberation is
"non attachment, fearlessness, and non-duality (no-self)"
They can still think and analyse like anybody else. It is still
a necessary function for daily living and to communicate the dharma
to others.
They still have the same physical appearance and needs (sleep,
toilet, sick, old age, death, etc) like any other humans. You will
not be able to tell by the looks. There are examples in the
scriptures where a person cannot recognise Shakyamuni was the
Buddha, or that certain people are Arhants, such as the example
given in the next post.