the Bernoulli integral can be expressed from the momentum equation after you apply some hypotheses, in this sense the original equation is more general.

You can see the Bernoulli integral also as a simplified form of the energy equation.

Yea sure.

In essence the answer is yes. You obtain the momentum equation applying Newton 2nd principle F=ma to a fluid element. If you discard viscous effects you end up with Euler equation: dp=-rhoVdV. This equation can be used for compressible and incompressible conditions. In the latter case rho is a constant. If you integrate between two points 1 - 2 of a streamline you obtain an equation in finite form that reads as Bernoulli equation: p2-p1 = 1/2 rho (V1^2-V2^2). The gravitational term rhogz (body forces) is not present here, which is rather common in gas dynamics.