If $xa=xb$ then $a=b$ We just defined the axioms of a group in our lecture notes on algebra, but silently assumed that the properties of the $=$ relation are known. A few lemmas after the definition of a group we prove that (if $G$ is a group):

$\forall x \in G, \forall a,b \in G: \ xa=xb \implies a=b$

by multiplying both sides by $x^{-1}$.
Now I wonder why we are allowed to do so. Is it a general property of equivalence relationships that if $a=b,c=d \implies ac=bd$ , or is it a group property?
Now I wonder why we are allowed to do so. Is it a general property of equivalence relationships that if $a=b,c=d \Rightarrow ac=bd$ , or is it a group property? 


It is property of binary operations. Binary operation on a set $G$ is defined as a function $G\times G\to G$. 

This means that the result $ac$ is uniquely determined by $a$ and $c$. Which is just rephrasing of the implication you wrote: $a=b \land c=d \Rightarrow ac=bd$.

(For functions we have $x=y$ $\Rightarrow$ $f(x)=f(y)$.)
Be: $ g * h = g * k$ ; then there is a $s$ in $ G $, so that: $g*s = e $; 

$s * (g * h) = (s*g) * h = e * h = h;  $

$s * (g * k) = (s*g) * k = e * k = k;$

And this gets you:  $ h = k; $ 

You can show in very similar manner that $ h = k $, if $ h * g = k * g; $