Yes, there is a general result, see Chapter 9 of Gnedenko-Kolmogorov book.

The theorem says that if $\xi_i$ are i.i.d with values in $\mathbb{Z}$ such that $$\text{gcd}\{s-s':\mathbb{P}(\xi_1=s)>0,\;\mathbb{P(\xi_1=s')>0}\}=1,$$ and $b_n^{-1}(T_n-a_n)$ converges in distribution to a stable law, then also the local limit theorem holds, namely, $$ b_n\mathbb{P}(T_n=k)-g(b_n^{-1}(k-a_n))\to 0 $$uniformly in $k$, where $g$ is the density of the stable law.

There is also a necessary and sufficient condition for the existence of $b_n$ and $a_n$ such that $b_n^{-1}(T_n-a_n)$ converges in distribution, see e. g. Feller's or Durrett's book.

Yes, there is a general result, see Chapter 9 of Gnedenko-Kolmogorov book.

The theorem says that if $\xi_i$ are i.i.d with values in $\mathbb{Z}$ such that $$\text{gcd}\{s-s':\mathbb{P}(\xi_1=s)>0,\;\mathbb{P}(\xi_1=s')>0\}=1,$$ and $b_n^{-1}(T_n-a_n)$ converges in distribution to a stable law, then also the local limit theorem holds, namely, $$ b_n\mathbb{P}(T_n=k)-g(b_n^{-1}(k-a_n))\to 0 $$uniformly in $k$, where $g$ is the density of the stable law.

There is also a necessary and sufficient condition for the existence of $b_n$ and $a_n$ such that $b_n^{-1}(T_n-a_n)$ converges in distribution, see e. g. Feller's or Durrett's book.