Many properties pass to hereditary sub-C*-algebras: simplicity, real rank zero, stable rank one, being an AF-algebra, purely infinite ... A useful result in this context is Brown's stabilization theorem, in Brown (1977) # Stable isomorphism of hereditary subalgebras of C-algebras [Pac. J. Math. 71]. It says that in a (say separable) C*-algebra, every hereditary sub-C*-algebra is stably isomorphic to the ideal it generates. Thus, every property of C*-algebras that passes to ideals and is preserved under stable isomorphism will pass to hereditary sub-C*-algebras of separable C*-algebras.

Yes, given a C*-algebra A and a hereditary sub-C*-algebra B of A, the primitive ideal space of B, Prim(B) is (homeomorphic to) an open subset of Prim(A). Let's assume that all C*-algebras are nonzero. Then A is simple if and only if Prim(A) is the one-point space. An open subset of the one-point space contains (at most) one point, therefore B is simple.

The spectrum of a nontrivial projection (i.e., nonzero and not the unit) is disconnected. Thus, a unital C*-algebra with the property you consider cannot contain any nontrivial projections. Conversely, if a C*-algebra contains a normal element with disconnected spectrum, then it contains a nontrivial projection. Thus, in a unital, simple C*-algebras without nontrivial projections (such as the Jiang-Su algebra), at least the spectrum of every normal element is connected.

Thanks, I realize that my question was imprecise. I will edit it. What I want to know is: Can $\|A\|_p$ be constant on some non-trivial interval, yet not globally constant?

Do you have a proof in mind? The inequality is not true in general, for example for the identity matrix.