$\newcommand\si\sigma$Note that 
$$\si(T)=\frac1T\,\int_0^T dt\,s(t).$$
Take any $L>\limsup s(T)$ and then take any real $A>0$ such that $s(t)\le L$ for all real $t>A$. Then 
$$\limsup\si(T)\le\limsup\frac1T\,\int_0^A dt\,s(t)+\limsup\frac1T\,\int_A^T dt\,s(t)\le0+L=L,$$
for any $L>\limsup s(T)$. 

So, the answer to your first question is yes.