If my brother had a child with immunodeficiency at birth, should I get tested to see if my children will have it?

No. It all depends on the type of immunodeficiency but most infant immunodeficiencies are secondary to a mutation. Do other members of the family suffer from the same? If so, yes you all need to be tested. Your question is best answered by a genetic counselor.
Depends on diagnosis. Chances of having children with a genetic condition depend on the specific genetic diagnosis. If the immunodeficient child was a boy, and the condition is x-linked, (gene defective is in chromsome x), then it is highly unlikely that the father has a similar gene defect. In all other scenarios most specific diagnosis is necessary for further counseling.