For this equation, does $f(−x)=f(x)$ or does $f(−x)=−f(x)$?

Substitute $−x$ for $x$ and simplify.

$y=\frac{2}{3}(-x)+1$

$y=-\frac{2}{3}x+1$

This result is not the same as the original, so it's not even, and it's not equal to the opposite, so it's not odd. The opposite of the equation would be

$-y=-\frac{2}{3}x-1$

For this equation, does $f(−x)=f(x)$ or does $f(−x)=−f(x)$?

$y=(−x+2)^2=x^2−4x+4$

Since $(x+2)^2=x^2+4x+4$ and $−(x+2)^2=−x^2−4x−4$ this equation is neither even nor odd.

See parts (a) and (b).

Yes, $f(−x)=f(x)$, making this even.