ihwip, the answer to your original question is that in principle, yes, individual quarks annihilate individual antiquarks when a proton and an antiproton collide. In practice, by the time the two are close enough for any of the constituent quarks to interact, the forces pulling together the rest of the quarks are so strong that there's no way for them not to come in contact and annihilate each other.
Yes, annihilation reactions do produce neutrinos in addition to photons. This is actually one reason antimatter as a power source is impractical - the neutrinos carry away a lot of the energy, making it very inefficient.
As for your question about quarks really existing or not, you should read the thread on the strong force in this forum. Because of the odd properties of the strong force, quarks are never observed as free particles, but we do have good reason to believe they exist.