second to last question.
Prove that lim_{x to x_0} f(x) = L if and only if lim_{x to 0} f(x+x_0) = L. Assume that L is finite.
That's pretty straight forward. In terms of the definitions of those limits you want to prove that
"Given $\displaystyle \epsilon> 0$ there exist $\displaystyle \delta> 0$ such that if $\displaystyle |x-x_0|< \delta$ then $\displaystyle |f(x)- L|< \epsilon$"
if and only if
"Given $\displaystyle \epsilon> 0$ there exist $\displaystyle \delta> 0$ such that if $\displaystyle |y|< \delta$ then $\displaystyle |f(x_0+ y)- L|< \epsilon$.
(I've changed "x" to "y" in the second part so as not to confuse the two uses of "x".)
Comparing those two, they will be the same if $\displaystyle x_0+ y= x$. In other words, let $\displaystyle y= x- x_0$.