Let $X$ be a stratified complex algebraic variety with smooth strata $U$, its inclusion is $j$ and $L$ is a local system on $U$. All the functors are derived if necessary.
Question 1 : Is it true that $j_*L[d]$ is perverse ?
What I tried : I can prove it when $D := X \backslash U$ is a normal crossing divisor but I'm not sure about the general case.
Question 2 : What is the simplest example where $^pj_{!*}$ is different from $^pj_!$ and $^pj_*$ ?
What I tried : If $X = (\Bbb A^1,0)$ , then $j_!L[1] = j_{!*}L[1]$ is just the extension by zero of $L$ in degree $-1$ and there is a short exact sequence $ 0 \to j_!L[1] \to j_*L[1] \to W \to 0$ where $W$ is a skyscraper sheaf supported on the origin.
Q1. If $p$ is the top perversity function, then $Rj_∗L[d]$ is a $p-$perverse sheaf. Dually, if $p$ sis zero perversity function, then $Rj_!L[d]$ is a perverse sheaf. I guess they are silly perversities. A generalization of what you tried could be the following : Let $j:U→X$ be an open affine immersion then thhe functor $Rj_∗$ takes perverse sheaf to perverse sheaf. This is corollary 4.1.10 in BBD, Faisceaux Perverse.
Q2. A simple where one could study the difference is when $X$ is a complex algebraic variety with isolated singularities. As the singularities get worse, the explicit computation becomes difficult. Maybe it would be best to compute with middle perversity in the case of isolated singularities. This(https://www.unirioja.es/cu/luhernan/jornadasmchagt/files/3joana_cirici_beamer.pdf) has the calculation I mentioned on page 4 or 5.