$f$ is integrable on $[a, b]$ and $F(x) = \int_a^x f(t) \, dt$. If $F$ is differentiable at $x_0$ is it always true that $F'(x_0) = f(x_0)$?

Hint: Consider a function $f$ which is the zero function except at a single point in the interval.

No, it is not. A counterexample is the function $f:[0,2]\to \mathbb{R}$ defined by $$ f(x):=\begin{cases} 1,& x=1\\ 0,& \text{ otherwise } \end{cases} $$

Then $F(x):=\int_0^x f(t) \mathop{}\!dt=0$ but $F'(1)\neq f(1)$.

The best approach to the problem is to understand the proof of Fundamental Theorem of Calculus. If you understand the proof well you should note that the proof is actually about this more general version:

Theorem: Let $f$ be Riemann integrable on $[a, b] $ and $F(x) =\int_{a} ^{x} f(t) \, dt$. Let $c\in[a, b] $ be such that one sided limit $f(c+) =\lim_{x\to c^{+}} f(x) $ exists. Then the right derivative of $F$ at $c$ exists and equals $f(c+) $. A similar statement holds for $f(c-) $ and left derivative of $F$ at $c$.

From the above it follows that if $L=\lim_{x\to c} f(x) $ exists then $F'(c) $ exists and equals $L$. But then this limit $L$ does not necessarily equal $f(c) $ (in other words $f$ may have a removable discontinuity at $c$) and then $F'(c) \neq f(c) $.

The above theorem also shows that if $f$ has jump discontinuity then $F$ is not differentiable at $c$ (left and right limits of $f$ are different and hence left and right derivative of $F$ are different).

Another more curious example is when $f$ has essential (oscillatory) discontinuity at $c$ and $F$ is differentiable at $c$. This is possible as exhibited by the function $F(x) =\int_{0}^{x}\sin(1/t)\,dt$. It can be proved with some effort that $F'(0)=0$.

One should also observe that the definition of $F$ as Riemann integral of $f$ over $[a, x] $ involves the behavior of $f$ in an interval. Changing the values of $f$ at a finite number of points does not affect the integral and hence does not affect $F$ and hence one should not feel surprised that properties of $F$ are not really dependent on values of $f$ at specific points. Thus one should not expect $F'(c) =f(c) $ in general. This happens in a very specific case when $f$ is continuous at $c$ otherwise this is not guaranteed.