The correct answer to the title: No.
Answers to your questions:
- 저는 저 자신이 무서워요 is like "I am afraid of myself"; 저는 저 자신을 무서워해요, "I fear myself." They are semantically (almost) the same, but Google search results suggest that the former construction is preferred when the subject is the first person.
Those two adjectives, 무섭다 and 슬프다, may be difficult to understand because they describe how the topic makes you feel and how the subject feels (Please note that not all other emotional adjectives describe both). As for the former, the following are examples:
사자는 무서워요. Lions are scary.
이 노래는 슬퍼요. This song is sad.
The latter—description of how the subject feels—is relevant to your example sentences; however, there is an important limitation for the emotional adjectives. When you use declarative sentences to describe the subject's emotions, the subject must be the first person, although there is an exception in literature (I explained the details here). For interrogative sentences, the subject must be the second person. These examples show how 무섭다 and 슬프다 are usually used as the emotional adjectives:
(저는) 무서워요. I am frightened.
(저는) 사자가 무서워요. I am frightened of lions.
(저는) 슬퍼요. I am sad.
(저는) 이 노래가 슬퍼요. I am sad about this song.
무서우세요? Are you frightened?
사자가 무서우세요? Are you frightened of lions?
슬프세요? Are you sad?
이 노래가 슬프세요? Are you sad about this song?
The verbs, 무서워하다 (무섭다 + -어하다) and 슬퍼하다 (슬프다 + -어하다), do not have the limitation mentioned above: They can be used with the first-, second-, and third-person subjects. One thing you should keep in mind is that 무서워하다 is transitive whereas 슬퍼하다 is intransitive and transitive.
저는 사자를 무서워해요. I fear lions.
저는 이 노래를 슬퍼해요. I feel sad about this song.
너무 슬퍼하지는 마세요. Do not feel too sad.
왜 사자를 무서워하세요? Why do you fear lions?
왜 슬퍼하세요? Why do you feel sad?
친구들은 사자를 무서워해요. My friends fear lions.
어머니께서 이 노래를 슬퍼하세요. My mother feels sad about this song.