Undergraduate / how is this intro? ("have 13 ships") [7]

Some suggestions:

Comma splice in an otherwise solid opening. Do what Kevin says =]

"He held the people of Korea even closer than his life and he embraced them..." => "He held his people even closer than his life. He embraced them..."

("He welcomed... He held... He embraced..." syntax patterning)

"Armor or rags..." => "Whether they were in armor or in rags"

"With weapon hidden" => "With a weapon hidden"? "With his weapons hidden"? "With weapons hidden"?

Undergraduate / how is this intro? ("have 13 ships") [7]

"the robot was unleashed at competition day" => "the robot was unleashed on competition day"

"Yi's guidance produced a robot whose..." => I think you should insert yourself in that sentence somehow. The reader should see more of YOU (in comparison to Yi Soon Shin)

"both piano and research paper..." => this part got a bit sketchy for me, particularly because you have not mentioned pianos and research papers prior to this.

Undergraduate / MIT Essay (End of the World) [13]

"Vader...Valli... Vallance"
So... you flipped past your guy... then back? I don't know if it was intentional...

Undergraduate / "First Impressions are Everything"; NYU Short Answers- grammar errors? [5]

"First Impressions are Everything" => I don't think the uppercase is necessary.

"and one mother never hesitated to remind me" => "and one that mother has never hesitated to remind me of"

"I was given a set of rules to follow; ways to behave, look, act, and speak." => "I was given a set of rules to follow: ways to behave, look, act, and speak."

"a beggar snatching a 10 dollar bill out of my hand" =>"a beggar snatched a 10 dollar bill out of my hand"