The following might be well known and/or embarrassingly simple. 
The motivation is the well known result by Auslander that the projective dimension of the Jacobson radical equal the global dimension minus one.
Let $A$ be a finite dimensional algebra with Jacboson radical $J$ and radical series $0=J^n \subseteq J^{n-1} \subseteq ... \subseteq J$.

Are the following true and/or well known:

1. Is the global dimension of $A$ equal to the injective dimension of $J$?

2.Is the sequence $id(J^i)$ of injective dimensions of $J^i$ weakly decreasing for algebras with finite global dimension (or general algebras)?

3. Is the sequence $pd(J^i)$ of projective dimensions of $J^i$ weakly decreasing for algebras with finite global dimension (or general algebras)?

I did tests for some quiver algebras and found no counter example, but those algebras had rather special properties. 
For example I found a proof of 1. for higher Auslander algebras.