The Cutting Plane View is an interesting area of TBC. It was created to facilitate the ability to slice a TBC data set e.g. a Point Cloud and to allow you to draw in the plane of the cutting plan while using the sliced data (a surface or a point cloud etc.) as a reference.
The thing to know about the Cutting Plane is that the XYZ axes of the Cutting Plane are a UCS (User Coordinate System) and are defined where the X and Y axes are the Across and Up/Down axes of the view with Z being in and out of the plane. When you draw in the cutting plane view, the data created is stored in the TBC Database as coordinates in the Cutting Planes UCS and while they can be seen and viewed in any view, they are in the different coordinate system - if you click on an object created in this way you will find that they have a UCS defined and that the coordinates of the data do not relate at all to the real world coordinates of your project.
When you create a cutting plane view you define the axes of the plane when you select the method of plane creation. Then you define the depth of the cutting plane and that will reveal more or less data either side of the cutting plane in the view.
When you draw in the cutting plane view, a 2D object like a Polyline will be drawn by default on the cutting plane itself unless you tell it that it has an elevation (offset to the cutting plane) that is other than undefined or 0 which is on the plane. When you draw an object like a linestring which is a 3D object and can have variable Z and you are using snaps say in a point cloud to extract locations, the linestring can criss cross the cutting plane depending on which points you snap to in the cutting plane. So if for example you are drawing a Building Elevation or a cross section in the cutting plane using the point cloud as a reference, use the polyline command rather than linestring to draw the linework in the plane itself.
Commands like Change Elevation when used in the Plan view change the true elevation of the point perpendicular to a horizontal plane, but when used in the cutting plane view will move the objects in the Z axis of the cutting plane i.e. perpendicular to the defined plane.
When you have a project that has some objects defined in a real world coordinate system and others defined in Cutting Plane Views that have their own UCS, you cannot combine, join, trim or extend one type of object to the other type of object because they have separate coordinate systems. If you want to combine the data then you can use the Convert to Linestring command to remove the UCS data from the Cutting Plane created objects which converts them into real world coordinate objects so that you can use them alongside all of the other data in the project. Doing this will remove the connection to the Cutting Plane view in which they were defined, so if you need to retain that connection you may want to copy the existing objects and then convert the copy objects leaving the originals behind.
The cutting plane tool is an awesome tool for many things, and can really help with some complex computations in TBC.
Check it out if you have not already done so.