Learn English – “We asked because of what she laughed” – why is the sentence wrong

The first one is a grammatical question. The second one reads more like a title than a question; it could be used as a header for a paragraph.

So, as an essay, you might see it like this:

How can one live happily? By learning to be content with what you have, and making the most of every day. A spirit of thankfulness helps as well.

or:

How one can live happily

We can have a happier life when we learn to be be content with what we have, and make the most of every day. Keeping a thankful spirit helps, too.

In a dialog, we might start with the question form:

"Dad, how can one live happily?"
"Well, son, we can be happy by learning to be content with what we have, and making the most of every day. Staying thankful can help, too."

B is wrong because "to be receiving" is a present tense and "won for its author" is something that has occurred in the past.

C is wrong because of ambiguity due to a risk that the interpretation could be possessive. To me, "the distinction of the first African American" is equivalent to "the first African American's distinction." "The distinction of being ..." cannot be confused like that, though.

D has a tense issue as @TRomano says. Had been implies there was, or could be someone else who could have later been another "first", which doesn't make sense.

E is wrong because was to be talks about an event that "will" happen (in the past) after another event (also in the past) - of course it's all in the past, but that's the idea. However, the sentence, when simiplified, says she X won Y a distinction - so this is a single event, not two events.