I am trying to check if these following inequalities are true
$$ \left| \sup_{0 < s < 1} (f_s + g_s) \right| \leq \left| \sup_{0 < s < 1} f_s \right| + \left| \sup_{0 < s < 1} g_s \right| $$
and $$ \left| \sup_{0 < s < 1} \int_0^s f_u \ d \mu_u \right| \leq \left| \sup_{0 < s < 1/2} \int_0^s f_u \ d \mu_u \right| + \left| \sup_{1/2 < s < 1} \int_{1/2}^s f_u \ d \mu_u \right|. $$
I think both inequalities are true in general. I argue about them as follows: assume that the maximum is at $s^*$, then
$$ \left| \sup_{0 < s < 1} (f_s + g_s) \right| = \left| f_{s^*} + g_{s^*} \right| \leq \left| f_{s^*} \right| + \left| g_{s^*} \right| \leq \left| \sup_{0 < s < 1} f_{s} \right| + \left| \sup_{0 < s < 1} g_{s} \right| $$
I can also argue for the second inequality using a similar idea by considering the two cases: when $s^*$ is in $[0,1/2]$ and when it's in $[1/2,1]$. However, I don't think this proof is correct in general, since such an $s^*$ might not exist.
How would I go about proving these inequalities in general?
Suppose $f_s = -1-s$ and $g_s = -1+s$ on $(0,1)$. Then $f_s + g_s = -2$, so
$$|\sup_{s\in(0,1)}( f_s + g_s)| =2 > 1 + 0 =|\sup_{s\in(0,1)} (-1-s)| + |\sup_{s\in(0,1)} (-1+s)| $$
Remark: it is not just the inability to achieve the supremums that is important; the next step that fails in your attempt is $$ \left| f_{s^*} \right| + \left| g_{s^*} \right| \leq \left| \sup_{0 < s < 1} f_{s} \right| + \left| \sup_{0 < s < 1} g_{s} \right|$$ and to violate this I used functions that can take negative values and the fact that $\sup f = -\inf (-f)$.
With regards to the second inequality, you didn't choose the measure $\mu$ so let me choose the Dirac mass at 1/2. Then $RHS = 0$ but LHS could be positive, by choosing any $f$ with $f(1/2)>0$.