Definiteness of quadratic form subject to linear constraint

Question

I have three questions related to the definition given in the image above:

1. Why do we only have a restriction $q\neq0$? Shouldn't we place the same restriction on p as well?

2. Is there a condition for a quadratic semidefinite form subject to a linear constraint?

3. This may be related to the second question: what happens if the determinant is zero?