The redshift caused by gravity is called gravitational redshift, which is different than the better known cosmological redshift caused by the expansion of space itself. To answer your first question, yes, gravity wells do create their own redshift! For example, a photon leaving the surface of, say, a white dwarf star will lose energy as it climbs out of the gravitational potential well. As the light loses energy, it will decrease in frequency and be redshifted when observed. Moreover, gravitational redshift is only significant for massive and compact objects (black holes, neutron stars, white dwarfs) and not really for the sun since gas motions near the surface of the sun cause a Doppler shift in the frequency of departing light that is larger than the gravitational redshift.
I’ll refrain from answering your second question since the posts above answered it well enough!
They estimate the depth of the gravity well. We sit in one ourselves so this can be taken into account as well. It doesn’t matter much. At distances where this is a large effect the random motion of galaxies is still important. At distances where you get nice measurements the redshift is so large the gravity wells don’t have a large impact any more.
Technically yes, but other effects end up being much larger - namely the expansion of space itself. Also, because black holes do not emit light, they end up not having much net effect on photons, as much of the blueshift cause while approaching is lost after it passes the black hole. The same is true for smaller gravity wells.
To jump on to this with a general relativity related question I've been pondering for a while.
When we're tracking the movements of probes and spacecraft as they move through the solar system, do we have to account for the effects of general (and possibly special) relativity in maintaining accurate knowledge of their positions and to manage maintaining a signal with them?
With the Juno probe for example. Does it's movement *away* from the Sun, or the Earth for that matter have to be factored into calculations which tell us exactly where it is, and what exact time its transmissions are sent? Similarly when it approaches a large body like Jupiter?