I am not a mathematician, so I have very little experience in this area, but from what I've understood by reading online, there are ways to assign certain values to some divergent series, and there are rigorous methods for doing so.
However, sometimes there are easy ways to find what those values are through simple multiplying, subtracting equations, rearranging the terms etc. For example, in this sum:
1 + x + x^2 + x^3 +... can we just say that it equals some number S for any x, even though we know that it doesn't always converge, and then go ahead and easily prove that S = 1/(1-x) for any x ≠ 1, as long as we accept that the answers we get when x ≤ -1 or x ≥ 1 are the regularized version of that divergent series?
In other words, if we can say that a divergent series is equal to a constant and then manipulate the equation so that we find what that constant is, is that considered mathematically correct and accepted or is the value decided through much more complicated and rigorous ways?
The situation is nuanced. That is, the manipulations that you describe implicitly assume certain conditions and properties hold. For example, suppose that you have some operator on infinite sums, denoted by $V$, such that $V(1+x+x^2+\dots)=:S$ gives the "value" of the sum. Then you need to assume that $$V(x(1+x+x^2+\dots)) = xV(1+x+x^2+\dots)=xS, \tag{1}$$ and also $$x(1+x+x^2+\dots)=x+x^2+\dots,\tag{2}$$ and also $$S=V(1+(x+x^2+\dots))=1+V(x+x^2+\dots),\tag{3}$$ and given these you get $\; S=1+xS\;$ and finally, assuming $x\neq1,\; S=1/(1-x)$. Given all these assumptions, the reasoning is valid. Coming up with a noncontradictory $V()$ is highly nontrivial.