Relation between best case and average case complexity
I think your definitions of $T_{\text{best}}$, $T_{\text{worst}}$, and $T_{\text{avg}}$ could use some clarification. I'll proceed assuming the correspond to best-, worst-, and average-case runtimes [1].

The first claim is false. The problem is that $f$ could dominate $T_{\text{best}}$ at an arbitrary rate. So we cannot make any claims about $f$'s status as an asymptotic lower bound of $T_{\text{avg}}$.

The second claim is true. Hint: prove $T_{\text{best}} = O(T_{\text{avg}})$, then use transitivity of $O$ to conclude the result.

[1] https://en.wikipedia.org/wiki/Best,_worst_and_average_case