I'm not sure your original question was on topic here at MSE, since it was a philosophy question rather than a mathematical question. However, your edit gives two clear mathematical questions I can answer.
As I understand them, they are
- Why is a negative times a negative equal to a positive?
- Is $-1$ a number, or is it shorthand for $0-1$?
1 has probably been posted many times on this site. Here is what I believe is the canonical version of the question. While omitting any justifications, since I'm sure you can find them given on that question, the short answer for why a negative times a negative is equal to a positive is that we can prove it from the axioms we've chosen for our mathematical system.
My response to 2 is that implicit in this question is a false dichotomy. You're implying by this question that $0-1$ isn't a number.
What are numbers though?
To answer this question though, we'd need to define what we mean by number, however there is no definition of number in mathematics. Instead it is a fuzzy word, and individual mathematicians interpret the word number differently.
Examples of objects that some might consider numbers (from roughly least controversial to most controversial)
- The integers, rationals, reals, the complex field
- The finite fields (not characteristic 0),
- The quaternions (not commutative),
- $k[x]$ for $k$ a field (A UFD, even a PID, but they're polynomial rings, at this point they are more rings of functions rather than rings of numbers, depending on who you might ask. Algebraic geometry complicates this distinction though.)
Nonetheless, almost everyone considers the integers to be numbers.
What are the integers?
Now we need to be clear about what the integers are. Almost everyone works within the axiomatic framework of ZFC, and we can construct an object we call the integers based on these axioms.
This object has certain properties subject to certain axioms. One of those is that for every integer, $n$, there is another integer $-n$ with the property that $n+(-n)=(-n)+n=0$. There is also a subset of the integers that we call the positive integers. The integers $-n$ where $n$ is positive are the negative integers.
What is the point of this explanation?
My point is that all of mathematics is a linguistic game, from sets to the positive integers to the negative integers to the complex numbers. None of it is real, or all of it is real (depending on your perspective).
Since I like to think of money as real, I also like to think of mathematical objects as real. Not in a truly platonic sense of thinking that somewhere out in the physical cosmos, there is a literal object that is "the integers," and that we are interacting with this magical entity, but rather I think the integers are real in a more pragmatic sense. There's no sense going around constantly reminding myself that money is a social construct, when if I run out of that particular social construct I will suffer real consequences.
In the same way, there are consequences to ignoring mathematics. I'm sure there are better examples, but one example is all the cranks that go around spending their time trying to square the circle, or double the cube with a straightedge and compass. It's a waste of time. More relevant perhaps, you can say that negative numbers are just social constructs all you like, but society uses them (for everything from physics to finance), and they don't go away if you pretend they don't exist.
Original answer below
While I'm not entirely sure this is on topic for MSE, since it's more a question of philosophy than of mathematics, I can't resist adding an answer.
Generally speaking I agree with J.G.'s answer (+1) and Don Thousand's comment on it.
I particularly like J.G.'s comment that negative charges and positive charges exist and cancel each other out (kinda anyway), so we have physical examples of things that negative numbers help us count.
Thus I'll leave your first question be, and address what you've written after.
To quote you,
Is there really anything more to the minus sign than a linguistic convention or more to the rule that $(-2)\times (-2) = 4$ than a linguistic stipulation (rule of the game)?
I think you're taking a sort of reductive view of things. You're asking, are negative numbers real, or are they just a linguistic game?
I would answer that they are both real and a linguistic game. As is logic, money, the color blue, and many other social or mental constructs. Social, linguistic, and mental constructs are real. We can detect their existence through the behaviors they cause us to produce. Not only are they real, they are valuable and effective tools that allow us to better our lives.
Negative numbers may be "merely" a social construct, but that in no way negates their reality or utility.