Q1: Yes. Since $f'_n$ is bounded in $L^p$, a suitable subsequence will converge $f_n'\to g$ weakly. Then also $$ f(x) =\lim f_n(x) = \lim \left( f_n(a)+\int_a^x f'_n(t)\, dt \right) = f(a) + \int_a^x g(t)\, dt , $$ initially for almost all $x$, but then in fact for all $x$ by continuity.

Q2: No. For example $f(x)=|x|^{-\alpha}$ has a derivative $f'\in L^p$ for $p<1/\alpha$, so can easily be approximated by functions with bounded $L^p$ derivatives.

Q1: Yes. Since $f'_n$ is bounded in $L^p$, a suitable subsequence will converge $f_n'\to g$ weakly. Then also $$ f(x) =\lim f_n(x) = \lim \left( f_n(a)+\int_a^x f'_n(t)\, dt \right) = f(a) + \int_a^x g(t)\, dt , $$ so $f$ is absolutely continuous (with derivative in $L^p$).

Q2: No. For example $f(x)=|x|^{\alpha}$ has a derivative $f'\in L^p$ for $p<1/(1-\alpha)$, so can easily be approximated by functions with bounded $L^p$ derivatives.