If $(v_n)_n$ is a sequence of Lipschitzian functions such that $$v_n(t)\to v(t) \quad\mbox{ uniformly in } [0, T],$$ is $v$ still a Lipschitzian function?
Moreover, it can be said that $$|v_n(t)|\to |v(t)| \quad\mbox{ uniformly in $[0, T]$, too }?$$
Thank you in advance!
Let me start with your second question.
Since $\bigl||v_n(t)|-|v_n(t)|\bigr|\leq |v_n(t)-v(t)|$ it is clear that $|v_n|$ uniformly converges to $|v|$ in $[0,T]$.
For what concerns the second question the answer is no, unless you have some equiboundedness on the Lipschitz constants. Take any continuous function $v$ on $[0,T]$ which is not Lipschitz and define $v_n(t):=\inf_{s\in[0,1]}\bigl(v(s)+n|t-s|\bigr)$. It is easy to see that $v_n$ is Lipschitz for every $n$ and $v_n$ converges to $v$ uniformly in $[0,1]$ as a consequence of Dini's lemma, however the limit is not Lipschitz by construction. If you take the sequence $v_n$ to be equi-Lipschitz ($\text{Lip}(v_n)\leq M$ for every $n$) and uniformly converging to some $v$ then you can try to show that the limit is actually Lipschitz as an exercise.