Suppose I have a strongly measurable mapping $f:[0,T]\to X$, $X$ a separable Banach space, which is Bochner interable, such that $$ \int_0^T \|f(t)\|dt < \infty. $$ Now I want to define a mapping $g$, which is, effectively, $g(t) = \int_0^t f(s)ds$. I believe I can rigorously construct this as $$ g(t) = \int_0^T 1_{[0,t]}(s)f(s)ds, $$ as $1_{[0,t]}(s)f(s)$ will also be Bochner integrable.
My real question is, can I now get away with manipulations like $$ g(t) - g(t') =\int_0^T 1_{(t',t]}(s)f(s)ds = \int_{t'}^t f(s) ds, \quad t' \leq t, $$ where I can treat the integral in a ``classical'' way. If yes, what justification is needed? If not, how can I understand what fails?
When thinking about integration on exotic spaces, as a general rule, it's "easy" to deal with exotic ranges and "harder" to deal with exotic domains. In particular, integrating maps from $\mathbb{R}$ to $X$ is not really any more difficult than integrating a function from $\mathbb{R}$ to $\mathbb{R}$ (the tricky part in either case is defining the Lebesgue measure on the domain).
A nice trick to reduce questions about $X$ to questions about $\mathbb{R}$ is to use the Hahn-Banach theorem. For your proposed manipulation, this goes as follows:
For any functional $\lambda$ on $X$, we have $$\lambda(g(t)-g(t')) = \lambda \left(\int_0^t f(s) ds - \int_0^{t'} f(s) ds\right) = \left(\int_0^t \lambda(f(s)) ds - \int_0^{t'} \lambda(f(s)) ds\right)$$ Now the point is that $\lambda(f(s))$ is a function $\mathbb{R}\rightarrow \mathbb{R}$, so your usual calculus rules apply. Then you have that $\lambda(g(t)-g(t')) = \lambda \left(\int_{t'}^t f(s) ds\right)$. Now since this holds for any such $\lambda$, by Hahn-Banach we must have $g(t)-g(t') = \int_{t'}^t f(s) ds$
Similarly, you can transfer all the regular calculus rules for maps into $\mathbb{R}$ to maps into $X$.
(A good reference for Bochner integrals, etc. is Lang's "Real and Functional Analysis", where the Bochner integral is constructed in such a way as to make clear that there is really nothing special about integrating real functions as opposed to Banach-valued maps.)