would the gravitational wave energy (that was lost from the objects producing the gravitational wave) be deposited inside the black hole?
At least some of it would. If a gravitational wave is incident on a black hole, then by conservation of energy, the sum of the initial energy of the wave and mass-energy of the black hole is equal to the sum of whatever wave energy is scattered and the final mass-energy of the black hole. (This follows from conservation of the ADM mass, which is conserved in an asymptotically flat spacetime and includes radiation that escapes to future null infinity.)
Or would the gravitational wave simply pass through the black hole?
No, I don't think so. The only way for the black hole not to gain mass would be if 100% of the wave's energy were scattered, and none absorbed, but there is no way that's going to happen.
For instance, consider the (unrealistic) case where the wavelength is small compared to the Schwarzschild radius of the black hole. Then the wave's energy can be localized to a region as small as a wavelength, and therefore when some of this energy, in a region of this size, passes inside the event horizon, it is guaranteed to be absorbed.
The more realistic case would be one in which the wavelength is millions of times bigger than the size of the black hole. I can easily imagine that in a case like this, some much larger fraction $f$ of the wave's energy would be scattered, perhaps almost all of it, but this $f$ has to be a smooth function of the initial parameters, so I don't see how it can become exactly 1.
Finally do gravitional waves "red shift" or "blue shift" due to the gravity of another object?
Yes, because the standard results for gravitational Doppler shifts depend only on two assumptions: (1) the wave propagates at $c$, and (2) the equivalence principle holds. Both of these assumptions are valid here.