It's all about the subject. As pointed out by Ben Walker, you have actually given us two sentences with the same issue. What helps is to simplify the sentence to subject and verb, removing phrasing. "Which of the sentences is/are grammatically correct?" Which is the subject. A trick an English teacher taught me is to say "which [one]...?" This helps you see that your subject is singular and thus, you would use a singular verb (e.g., "is").
We use "one" because what you are choosing between are two singular items. If, however, you were choosing plural items, such as which M&M colors you wish to eat, your sentence structure would be plural. Using above trick, the question "Which M&M colors are you going to eat?" is "Which [ones] are you going to eat?"
Another little trick that can be helpful is to read the subject and verb aloud. "Which is...?" or "Which are...?" Stilll unsure, say outloud, "Which [one] is...?" or "Which [one] are...?" "One" is singular, so you use the singular verb "is". Or "Which [ones] is...?" or "Which [ones] are...?" "Ones" is plural, so you use the plural verb "are".
Same principle applies to the "Being egocentric and selfish..." sentences. Here, "egocentric" and "selfish" are each singular items combined together as a single unit (i.e., singular). So instead of reading it as "Being [blank and blank]", read it as "Being [blank]". Read aloud, "Being [blank] is dangerous...", "Being [blank] are dangerous..." The answer is "Being [blank] is dangerous." Singular subject uses a singular verb.
Whew, this is my first answer. Seeing the length of it, I do believe I need to work on how to clearly summarize a shorter answer! But I do hope something from the answer above may be of use to you.