Copying a portion of a LiDAR file and pasting it into the Project pane creates a table. If I use that table to create an image, the projection information is permanently damaged. I cannot restore the parent projection to the child dataset. This includes creating a Favorite projection setting that overwrites metrics.
This sounds like a series of procedural errors. You don't say precisely what you are doing so I cannot point to specific steps you might need to correct.
1. LiDAR data isn't an image, but a drawing of points. That means it is point data stored in a table that is visualized by a drawing. See the illustrations in the "Drawings and Tables" section of the Drawings topic.
2. The coordinate system in use by drawing's table is a property of the table, which you can see in the Properties dialog of the table (right-click the table in the project pane, choose Properties), as illustrated in the above Drawings topic. It's the FieldCoordSystem.Geom property.
3. When you copy points "from a LiDAR file" I presume you're copying points from the LiDAR drawing (that is the correct way). Pasting those points into the project creates a table.
4. If you want to create a drawing from that pasted table (note that you create a drawing, not an image), you have to say what coordinate system you want it to use. The easiest way to do that is to simply copy the value of the FieldCoordSystem.Geom property from the original LiDAR drawing's table, and then in the pasted table create a FieldCoordSystem.Geom property and paste the value from the original table into that. Takes but a few seconds and a couple of clicks. You can then create a drawing on that table (couple of clicks) and the correct coordinate system will be in use.
I just did that, using the sample Pentagon LiDAR data set that's used in examples and on the downloads page. Copy points from the drawing, paste into the Project pane as a table, copy and paste the FieldCoordSystem.Geom property's value into the new table and create a drawing. The pasting and adding the coord system and then creating the drawing takes a just a few clicks. Drag and drop the new drawing into a window showing the old drawing, and everything lines up perfectly.
Once you have a new LiDAR drawing that is a subset of the original data set, you can create images from that using whatever interpolation method you like.
If you have strong skills, you could create subsets from images, but to my mind that's not only the long way around asking for trouble if you overlook some detail, but it also loses accuracy since LiDAR data is vector point data, not pixel raster data.
One last thing, a minor detail although just about everybody ignores it: "the acronym stands for Structured Query Language."
Nope. It's not an acronym. See the Notes section of the Queries topic.