Back to indicating chuck and tailstock.
1) normally chucks are expected to be not much closer to zero than .003
2) No indicator should be expected to not react to gravity as it is rotated about a horizontal axis.
Use your indicator without rotating it about a horizontal axis, it will not be accurate if it is rotated about a horizontal axis. To verify this, mount a magnet base on the face of a chuck, put the indicator on the top of the chuck and rotate it 90º.
To align chuck and tail stock, a test bar, center drilled on one end and turned to one diameter for its length, should be used. Mount it in the chuck and check that it is concentric to the spindle, shim the bar or move the chuck until you have zero (or nearly so) runout at the chuck, then engage the live center with enough pressure to turn the center with the chuck, snug but not TIGHT. mounting the indicator on the saddle, and against the test bar, first check the top for parallel the ways. Once this is correct, check the side toward you from end to end. If it is not parallel, adjust the tail stock sideways and check again until there is no runout.
Without a test bar, it can be done by turning a piece between chuck and live center, but it's not as quickly accurate and takes a lot longer.