Partial functions are not encountered in many areas of mathematics. As you wrote, we can often just change the domain (to make a total function with the same graph and a smaller domain) or we can use a new "flag" value to indicate when the original function was not defined.
Computability theory is one of a few areas (unbounded linear operators are another) where partial functions are taken more seriously.
I think one reason computability theorists like the $\simeq$ notation is that
it reminds us that $F(x)\simeq y$ is really two claims: first, that the computation of $F(x)$ eventually terminates; second, that the resulting value is y.
Another notation that is sometimes used: "$F(x)\mathord{\downarrow}$" means that the computation of $F(x)$ eventually terminates - which we describe in words either as "$F(x)$ is defined" or "$F(x)$ converges" (or even "$F(x)$ halts"). The accompanying notation "$F(x)\mathbin{\mathord{\downarrow}\mathord{=}}y$ " means that $F(x)$ is defined and equal to $y$ - the same as $F(x) \simeq y$.
Having both claims visible helps remind us what the proof of $F(x) \simeq y$ needs to establish.
The notation also helps us keep in agreement with a common idiom in free logic, a logic that includes undefined terms. In many free logics, only things that are defined can be equal to anything - in that context, $F(x) = G(x)$ would imply that $F(x)$ and $G(x)$ are both defined and are equal, while the computability notation $F(x) \simeq G(x)$ says that either $F(x)$ and $G(x)$ are both undefined, or are (both defined and) equal.
There is also a pragmatic reason we don't come up with another symbol that means "not defined" so that we can redefine partial functions to be total, but taking this new value when they are not defined. Even if the original partial function was computable, the new "total" function may not be.
For a standard example, let $\phi_x$ be the computable function with program $x$, and let $f(x)$ be the partial computable function that, if $\phi_x(0)$ halts, returns the number of steps that were required for it to halt. So $f(x)$ is a partial computable function which is defined on the undecidable "halting set" $K = \{ x : \phi_x(0)\downarrow \}$. We can easily show there is no total computable function $g(x)$ that agrees with $f(x)$ - if there was, we could use $g$ to solve the decision problem for $K$. This is true if $g$ says "not defined" on inputs where $f$ was not defined, or if $g$ returns any other number it wants. Either way, we can use the output of $g(x)$ to tell whether $\phi_x(0)$ halts, by just running $\phi_x(0)$ for $g(x) +1 $ steps. (This means there is no algorithm at all - even a completely nonobvious one that ignores the definition of $f$ - that computes a total function extending $f$.)
Of course, we could artificially limit the domain of $f$ to $K$, so the restriction is a total function. But, because $K$ is not computable, this means we have no way to tell which numbers are in the domain, which leads us back to the same problems. It turns out to be easier to think of the domain as always $\mathbb{N}$, and handle the fact that a function may be partial with the $\simeq$ notation.
Best Answer
To refer to contradiction sometimes the symbol $$\Rightarrow\Leftarrow$$ however usually the word contradiction itself is used. But there is no symbol in mathematics for something absurd.
As you have rightly noted ! is used for factorial, negation and set theory. There you have it. If you want some more you might want to check out wikipedia.