The following versions sound correct to my (American) ear:
1a. The biggest fear people have is not being approved [for a credit card].
1b. The biggest fear people have is not being approved of.
3 . Elderly people have a desire to feel needed.
5 . Elderly people have a desire to be needed.
The following versions sound more natural to me:
1e. The biggest fear people have is that they will not be approved [for a credit card].
1f. The biggest fear people have is that they won't measure up [to society's standards].
3b. Elderly people want to feel needed.
5b. Elderly people want to be needed.
Notice that there is a difference in meaning between "approved" and "approved of". "Approved" implies "approved for <something>" like membership in a group, or permission to do something (like borrow money on a credit card). "Approved of" means "appreciated", "valued", or "loved".
Many people fear not being approved for a loan (such as a credit card), but it would be very strange for a person's biggest fear to be "not being approved for a credit card". I can imagine a sit-com (like Seinfeld) showing a neurotic character whose biggest fear is "not being approved for a credit card".
A common phrase in English is "fear of <plural noun>" or "fear of <gerund>". For example, "A person with acrophobia has a fear of heights", "She is afraid of cats", and "fear of flying".