Except that the number of dimensions is a global property of the entire space. You can’t have four dimensions here and another number of dimensions somewhere else.
The dimensionality is a count of the number of coordinates necessary to locate a point. It does not matter whether the coordinates are positive, negative, or zero. Let’s take the 2d plane. It takes two coordinates to locate a point. The coordinates themselves can be positive, negative, or zero. The space has two dimensions because it takes two coordinates to locate a point.
No. The extra coordinates are more like coordinates on a sphere, like latitude and longitude lines on the surface of the earth. If you measure the number of degrees from a particular longitude line and the number of degrees above the equator, you get two coordinates. These coordinates differ from coordinates in a plane in that they are cyclical. x + 360[sup]o[/sup] is the same coordinate as x. The dimensionality is the count of the coordinates. The nature of the coordinates does not matter.