1. It means your Milli-Q water is contaminated with particles. Zeta potential is a property of particulate matter in suspension and is meaningless for a pure liquid Check your system with a zeta potential standard.
2. Carry out a Stokes' law calculation. You'll have significant sedimentation above 10 microns for a particle above 2.6 g/cm3 density. Typically zeta potential is only important in systems < 10 microns as inertial forces dominate over van der Waals above this level. You can filter to 2 um or so and measure the ZP on the resulting particles (if any).
1. The zeta potential is the effective charge of whatever scattered light. If you measure a certain (and repeatable) negative zeta potential of a sample prepared in milliQ water then that means that the scattering objects - whatever they may be (sand, dust, contamination, additives,...) - show an effectively negative net charge. A good way to check for purity of milliQ water is to look at dynamic light scattering (DLS) and pay attention to the value for the scattering intensity.
2. While sedimentation starts to come in at a couple microns, it may still be possible to perform an electrophoretic light scattering zeta potential measurement at larger size: if the applied electric field is stronger than the sedimentation, measurements of particles up to 100 microns have been done. As indicated the use of zeta potetnial for stability prediction is of limited value in that regime. The precipitation may be the large particles themselves, and they will settle. To slightly expand the conditions to "make the measurements more reliable" you may consider modifying your sample viscosity (to slow down sedimentation, increase the viscosity) and/or sample density (the closer to density match of particles and dispersion medium, the slower the sedimentation), for example add sucrose or use D2O. However this will of course alter your overall sample conditions...