Under British/American common law, yes. Cuius est solum, eius est usque ad coelum et ad inferos—You owned a column to the center of the earth and infinitely into space. That doesn’t include mineral rights though, and since the invention of the air plane, air rights are now limited to only a few hundred feet typically.

Depends on the jurisdiction really, whether mineral and water rights are separable from the surface plot, and how “property rights” are defined and entitled. In the United States generally a property deed entitles you to exclusive use of the surface and soil. Surface water, groundwater, rocks and minerals beneath the soil (down tens of thousands of feet), and even air space, are wholly different sets of rights that may be deeded, traded, sold, or restricted. For instance in the western US (as opposed to the eastern states) surface water (creeks, streams, lakes) on your property may be entitled to a downstream user and is not automatically “yours” to use.