My goal right now is trying to build a new PC that will be as silent, and cool as possible using fans. Is the Lian Li 6070 Plus a good choice for this? I've also been interested in a Thermaltake Armor or Antec p180 but will gladly take other suggestions.
No, get the P180. Dual chamber cools better than the single chamber of the 6070 or Armor. It has silicon grommets, which I don't think your other two choices have, and the aluminum/plastic/aluminum sandwich with decrease vibrations transmitted to the case, and hold it airbone acoustic noise better due to the extra thickness of the walls. Aluminum cases tend to have more resonance due to the low density of the material, having a more audible "hum" than steel cases, or in this case, the P180.