Is there a single word for attitude of showing off one's weaknesses?
Solution 1:
A good word for this is malingerer. As Dictionary.com defines it:
to pretend illness, especially in order to shirk one's duty, avoid work, etc.
It is also used in the sense of exaggerating illness or disability. Merriam-Webster defines it this way:
to pretend or exaggerate incapacity or illness (as to avoid duty or work)
There's a whole Wikipedia article devoted to describing malingering, but most people will probably be familiar with the term from football/soccer. It describes when a player suddenly throws himself to the ground, faking an injury, in a pathetic attempt to draw a foul on another player.
Solution 2:
I suggest feign - verb -
1) to give a false appearance of : induce as a false impression
2) to assert as if true
http://www.merriam-webster.com/dictionary/feign
After the soccer player's fall, he feigned a serious leg injury to justify his poor performance.
Solution 3:
The sports player case depends on whether the injury is real or exaggerated. Ignoring real injuries could be termed stupid, or carrying on "gung ho"
Urban dictionary: gung ho, mandarin g0nghé .. soldier slang from 1940's meaning to be psyched out for war.
Note if the injury is faked, the sports player could be described as "taking a dive", meaning to try and lose deliberately, or to be "going for an oscar" meaning to act out a facade in the hope of getting positive referee intervention.
A word that comes to mind for the case of focusing on a disability in hope of reward is victim, used in contexts such as in "always playing the victim" or "playing the victim card". There's a great quote from the movie "Adams Family Values" after Amanda volunteers to take part in a life saving demonstration:
Amanda: I'll be the victim!
Wednesday: All your life.
You could also use the phrase "to milk something for all it's worth" in the context of a questionable display of weaknesses.
For single word answers on opposides of the spectrum I would suggest "gung ho" and "victim".