Would this statement, $\forall k \in \mathbb{R} $, be true or false? And are these interchangeable under all circumstances?
$$ \int f(x) dx + k= \int f(x) dx $$
Since we could assign $F(x)$ to either the LHS or RHS, and differentiating F(x) would both result with $f(x)$. However if we assign $I = \int f(x)dx$, then
$$ I = I + k \Rightarrow k = 0 $$ even though we already stated "for all k". I think my misunderstanding stems from notation and what they represent. Would appreciate any clarification on this as I could not find a direct take on this flexibility of antiderivatives, especially in such context of equality.
No, your statement doesn't hold in general, I think you're misunderstanding the notation a bit. For a differentiable function $F$ such that $F' = f$ we have that $$\int f(x) dx = F(x) + k$$
Notice that functions that differ in a constant have the same derivative, that's why we consider $k$ when writing the antiderivative. For example take $f(x) = x^2$ and $g(x) = x^2 + 1$, then $f'(x) = 2x = g'(x)$.