Do all galaxies radiate gravitational waves? What is the origin of these waves, the origin of the Galactic center? If it exists, do two galaxies warp together due to these waves, when they come closer?
Gravitation waves require a system with an oscillating quadrupole moment. There's a good description of what a quadrupole moment is here. The simplest example of a quadrupole is a rotating dumbell, and two stars rotating around each other have this geometry. In fact gravity waves have been inferred for binary pulsars.
In general galaxies are close to radially symmetric, so you would not expect the galaxy as a whole to radiate gravity waves. However galaxies have black holes at their centres, and if these black holes swallow stars there will be a burst of gravity radiation as the star spirals into the black hole. There will also be gravity waves from smaller black holes or indeed other dense objects like neutron stars scattered through out the galaxy.
I'm not sure what the last bit of your question is asking. In general gravity waves are barely detectable let alone likely to have any detrimental effect. You'd have to be very close to a gravity wave source before you'd even notice. The gravity wave causes a stretching and compressing force as it passes. You'd feel as if you were being alternately stretched then compressed at whatever frequency the gravity wave had. Gravity waves are unlikely to do much to two colliding galaxies.
Having said that, colliding galaxies could be a source of gravity waves. However the majority of any radiation would come from the central black holes swallowing stars as their orbits were disrupted, and eventually the merging of the central black holes.