If $A_n$ converges $A$ in Hausdorff distance, does this tell us anything about possible convergences (weak or otherwise) of subsequences of the indicator functions $\chi_{A_n}$ and $\chi_A$?
Because of convergence in measure would give us convergence strongly of the characteristic functions and I want to know if it can be expected here
No. Not even on the line (with the standard metric and measure).
Let $$A_n=\bigcup_{j=1}^n\left[\frac{2j-1}{2n},\frac{2j}{2n}\right]$$and $$A=[0,1].$$Then $$d_H(A_n,A)\to0$$but in fact $$\chi_{A_n}\to\frac12\chi_A$$in various weak topologies