I need my work checked(properties of operations) Let $x * y = |x + y|.$
$x * y = |x + y| = |y + x| = y * x,$ so $*$ is commutative.
$(x * y) * z = ||x + y| + z| = |x + |x + z|| =  x * (y * z),$ so $*$ associative.
$x * e = |x + e| = x,$ so $e = 0.$ Further, $e * x = |0 + x| = x.$ So, $*$ has an identity.
$x * x' = |x + x'| = e,$ so $x' = e - x.$ Further, $x' * x = |e - x + x| = e.$ So, every element has an inverse.

Let $x * y = |xy|.$
$x * y = |xy| = |yx| = y* x,$ so $*$ commutative. 
$(x * y) * z = |(xy)z| = |x(yz)| = x * (y * z),$ so $*$ associative. 
$x * e = |xe| = x,$ so, $e = 1$. Further, $e * x = |1x| = x$. So, $*$ has no identity.
$x * x' = |xx'| = e, $ so $x' = \frac ex$. Further, $x' * x = |\frac ex x| = e$. So, every element has an inverse.

Let $x * y = \sqrt{x^2 + y^2}.$ 
$x * y = \sqrt{x^2 + y^2} = \sqrt{y^2 + x^2} = y*x$, so $*$ is commutative.
$(x * y) * z = \sqrt{(\sqrt{x^2 + y^2})^2 + z^2} = \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (\sqrt{y^2 + z^2})^2} = x * (y * z),$ so $*$ is associative. 
$x * e = \sqrt{x^2 + e^2} = x,$ so $e = 0.$ Further, $e * x = \sqrt{0 + x^2} = x.$ So, $*$ has an identity.
$x * x' = \sqrt{x^2 + x'^2} = e,$ so $x' = \sqrt{e^2 - x^2} \text { and } x' = -\sqrt{e^2 - x^2}$. Further, $x' * x = \sqrt{(\sqrt{e^2 - x^2})^2 + x^2} = e.$ So, every element has an inverse. 

Let $x * y = x - y.$
$x * y = x - y \neq y - x,$ so $*$ is not commutative.
$(x * y) * z = ((x - y) - z) = (x - (y - z),$ so $*$ is not associative.
$x * e = x - e = x,$ so $e = 0$. Further, $e * x = 0 - x = -x.$ So, $*$ has no identity.
$x * x' = x - x' = e,$ so $x' = x - e$. Further, $x' * x = x - e - x = -e.$ So, not every element has an inverse.

Let $x * y = xy + 1.$
$(x * y) * z = xy+ 1  = 1 + xy,$ so $*$ is commutative.
$(x * y) * z = (xyz + z) + 1  = xyz + (z + 1),$ so $*$ is associative.
$x * e = xe + 1 = x, $ so $e = 0$. Further, $e * x = 0x + 1 = 1.$ So, $*$ has no identity.
$x * x' = xx' + 1 = e,$ so $x' = \frac {e - 1}{x}.$ Further, $x' * x = \frac {e - 1}{x} x  + 1 = e$, so every element has an inverse. 

Let $x * y = max\{x, y\}.$
$(x * y) * z = max\{max\{x, y\}, z\} = max\{max \text { } x, \{y , z\}\},$ so $*$ is associative.
$x * e = max\{x, e\} = x,$ so $e = min\{x, e\}.$ Futher, $e * x = max\{min\{x, e\}, x\} = x,$ so $*$ has an identity. 
$x * x' = max\{x, x'\} \neq e,$ so $x'$ is undefined. Not every element has an inverse.

For number 5: 
it is not associate. 
you did not check:
$(x∗y)∗z=x∗(y∗z)$
Also, you said "$xe+1=x$, so $e=0$" well if $e=0$ then we get $1=x$ but we need $x=x$ either way, you are right there is no identity but you did not provide the right reason.
This is important: If there is no identity then there is no inverse. So you contradicted yourself in the inverse part. 
Hopefully this helps!
