Are you doing this yourself? If not and you are getting the quartz from a dealer/installer, They will come and measure and then remove your existing countertop and replace with new one. You are paying for that service. It should not damage the cabinets as they take all that into consideration and as far as your tile goes, they may just install butted up to the tile rather than vice-versa. You would lose the bottom of the tiles based on thickness of counters. A professional will know what to do. just ask a lot of questions.