Is the salary received by a brother working in the army, where he is forced to shave his beard, salute the flag, and commit other forbidden acts, considered permissible (halal)?

If your brother's work is permissible, then his salary is lawful. The sin of the Sharia violations he is forced into falls upon the one who commands them and upon the doer if he is not compelled. This does not affect the lawfulness of his salary.