What does the notation of “1=/= 0” mean for “Let R be a ring with unity 1=/= 0”? Should I read it literally? Or does it mean that 
the multiplicative identity on the left hand side is not equal to the additive identity on the right hand side?

I think the answers are clear. Here is one more just to be explicit. We are concerned with commutative rings with a multiplicative identity.
Suppose I have a set $S=\{a,b,c\}$ with commutative operations satisfying $a=a+c=b+b=ab$, $b=a+a=b+c=bb$ , 
$c=a+b=c+c=ac=bc=cc.$ Then one way or another you can check the rules for a ring and verify that the structure is a ring with additive identity $c$ and multiplicative identity $b$. A shorter way to say that is $0=c$ and $1=b$.
Without the rule you mention $T=\{x\}$ would count as a ring with $x+x=xx=x.$ Since $a+x=a$ for everything in the singleton set $T,$ $x=0$, similarly $ax=a$ gives $x=1.$ 
It turns out that $T$ is the only thing which follows all the ring rules except the requirement  $0 \neq 1$. It is desireable to not have $T$ count as a ring. That uniqueness. Claim isnt that hard to prove but first you need some smaller results such as $0a=0$ . I do not recall for sure, but it might be cleanest to use $0\neq 1$ in an early proof.
