How to create a score for a SWOT analysis (strengths, weaknesses, opportunities, and threats)?
Let's suppose you have a variable $x_{old} \in [-1,1]$. If you would like to transform that to $ x_{new} \in [-2,2]$ then you could just apply the transformation formula you posted:

With $min_{old}=-1, max_{old}=1, min_{new}=-2, max_{new}=2$ you would get $$\frac{max_{new}-min_{new}}{max_{old}-min_{old}} (v-min_{old}) + min_{new} 
= \frac{2-(-2)}{1+1} (v+1) + (-2) = \frac{4}{2} (v+1) -2\\= 2v+2-2=2v$$

Now if you compare that to your formula

= (((strengths + opportunities) - (weaknesses + threats)) /
 ((strengths + opportunities) + (weaknesses + threats)) * 2


then you can see that it is exactly what you did here by multiplying with 2 at the very end. You transformed your first scale from $[-1,1]$ to $[-2,2]$. Therefore, if you want it to be in the interval $[0,10]$ just apply a different transformation at the very beginning when deriving the first scale:

$$\frac{max_{new}-min_{new}}{max_{old}-min_{old}} (v-min_{old}) + min_{new} 
= \frac{10-0}{1+1} (v+1) + 0 = 5v +5$$

Which means your calculation for the first scale needs to be

= (((strengths + opportunities) - (weaknesses + threats)) /
 ((strengths + opportunities) + (weaknesses + threats)) * 5 + 5


(instead of scaling it to $[-2,2]$).

Intuitively I like to think of it like this: you "stretch" the interval from $[-1,1]$ to $[-5,5]$ by multiplying with 5 ("making it 5 times larger"). And then, since you want it to be positive and start at $0$, you "shift" it "upwards" by adding 5.