Fundamentally, an equation is just a segment in a sentence, or sometimes a full sentence. For example, if I write:
Let $x$ be a number such that $x^2 = 4$, then $x < 4$.
This is a sentence, where I replaced some of the words with symbols. Of course, you could write the same using words:
Let $x$ be a number such that $x$ squared is four, then $x$ is less than four.
So when you see $x^2 = 4$, you should just think of it as the words “$x$ squared is four”. Is that a problem you can try to solve? Well not really. Is that a function? Well not really either. It's just a statement, like if I said “My car is blue”. In fact it doesn't even have to be true! I can say $1 = 2$ without any issue. It's a lie, but I can still say it, just like I could say my car is blue even though I don't own a car.
Depending on the surrounding sentence, it may become a problem, e.g. if you see “Find a number $x$ such that $x^2 = 4$”, then you have a problem you can try to solve! But the equation is not the problem — it requires the surrounding sentence.
Now, you might be confused because your school assigns you problems just by writing an equation. But such a problem would typically be associated with the words “solve the equation ...”, which is short-hand for “Find all possible numbers $x$ such that ...”.
As an example “Solve the equation $x^2 = 4$” is short-hand for
Find all numbers $x$ such that $x^2 = 4$.
Notice in particular that the equation really is part of the sentence; in fact the equality symbol is the verb!
Now, let's talk about $y = x + 1$ vs $y(x) = x + 1$.
You say that a function is something like $y(x) = x + 1$, but this is a misconception. It's just a statement about some things denoted by $x$ and $y$. Take the following example
Let $y$ be the function given by $y(t) = 2^t$, then we have $y(x) = x + 1$ when $x = 1$.
Notice how this sentence contains $y(x) = x + 1$, but the function $y$ is not at all the function that adds one. It's just a statement about $y$ and $x$, which in this case happens to be true. (because $2^1 = 1 + 1$)
Additionally, the equation $y(x) = x + 1$ is not the function. The function is just the thing we denote by $y$, and the equation is a statement about this function that gives us some information about it.
You say this:
In 'A' I understand better that $y$ is a function even though it gives me the feeling that it isn't really the function that is subtracting by 1, but instead its value when given an $x$, but maybe that does mean the same
Your feeling that $y(x)$ is the value when given $x$ is exactly correct! When I say $y(x)$ I am in fact talking about some number. If $x$ is another unknown number, I may not know which number $y(x)$ is, but I assure you that it is a number. If I want to talk about the function, I just say $y$.
Note that most people, including many teachers do not appreciate that there is an difference between $y(x)$ and $y$, but I find it unlikely you will get into trouble by doing it correctly.
Is $y(x) = y$ possible? Generally this is misuse of notation. The left side is a number and the right side is a function.
However you must unfortunately be prepared for people to misuse notation.
If someone says before-hand that $y$ is a function, and then writes this:
Is he wrong for not using the $y(x)$ notation?
Well, if $x$ is a number then yes, although I would like to introduce something you may find interesting: if $x$ was another function, then it could indeed be perfectly valid.
The idea is that you can think of think of functions as values you can manipulate just like you can with numbers. You are most likely to encounter this concept in the form of the function composition operator, which is written like this: $f \circ g$.
Basically what it means is: Given two functions $f$ and $g$, the expression $f \circ g$ represents a third function such that $(f \circ g)(x) = f(g(x))$ for every $x$. So this means it's treating functions like something you can do something with and that you can build new values by putting others together, just like $+$ does to two numbers.
(Note that people can't agree whether it should be $f(g(x))$ or $g(f(x))$.)
Now, it does make sense to ask if you can add two functions. It turns out that you can define this in a way that makes sense, namely if $f$ and $g$ are functions, then $f + g$ is a function such that given any number $x$ we will have $(f+g)(x) = f(x) + g(x)$.
I hope this helps you build some intuition for what equations and functions are, and feel free to ask any questions.