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#2




No. It is a strange thing you are asking about, but I don't think your conclusion is right.
First, I think you must mean E[(X+50)^1000]. Parentheses may not matter, but I assume this is what it meant. Then, suppose (as a counterexample showing the equality is not valid) X is always 2000. X+50 is always 2050, so E[(X+50)^1000] = 1000. But E(X^950) is 950. 
#3




In general, no.
Consider the formula for E[ X ^ d ] where S(y) = 1  F(y): Then: Do some algebra, then work out the formula for E[ X ^ (d  k)], and you'll find that
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