No. However, it is scratch resistant. In order to scratch any object, the item causing the scratch must be harder than the item being scratched. Since only a few things are harder than granite (quartz, silica and diamond), it is difficult to scratch. But remember, the un-glazed ceramics on the bottom of many kitchen items are full of silica.
It is best to use trivets. The heat created from your stove or your cooktop can not hurt the stone. However, a sudden thermal change from cool to hot could cause an air pocket within the stone to expand and “pop” out a piece of quartz.
No. Your countertops have been mechanically polished using diamond abrasives and water. The polish will not wear off. You can use polish if you want an extra sparkle in your counters, but it is not necessary.
In the case of granite (a stone made from silica), standard cleaners will not harm the stone. However, it will break down the formulation of the sealer that has been applied to your countertops. As for marble (a stone made from calcite) can easily be etched by many of the chemicals that are in standard cleaners.
Most importantly, your countertops have been sealed 1-2 times prior to completing your project. However, some stone may require additional applications down the road. Each type of stone differs in its porosity and how often it needs to be resealed (length of time will also vary depending upon the cleaners you use on your countertops). If your counters darken when they get wet, you know that it is time to reseal your counters. Resealing is a fairly simple wipe-on, wipe-off process and nothing to be concerned about.