This is how the fittings work:If I cut there, unscrew the nut on the left, how does the remaining piece of copper pipe that disappears inside the T through the nut come away with the nut, is it loose or somehow attached to the T?
Compression fitting isolating valve 15mm screwdriver turn [Public domain], by Mike1024 (Mike1024), from Wikimedia Commons
The nut and the olive (copper ring) both fit over the copper pipe. The pipe fits inside the valve / tee or whatever else.
When tightened, olive is compressed between the nut and the fitting, causing it to deform and grip the pipe, and at the same time seal against the tapered part of the fitting.
When the nut is undone, the pipe can be removed from the fitting, the olive will remain compressed onto the pipe.
The pipe is not threaded and does not fix to the fitting in any way - the only thing holding it in is the pressure of the nut against the olive.
If this type of fitting is not tightened enough, it will leak. If overtightened, the olive will be damaged and it will leak.
Both. If you undo both nuts, you will have a very short piece of copper pipe with two nuts on it, and an olive on each end which prevents you from removing the nuts.So this 'olive', is it underneath the nut that's screwed onto the T? Or under neath the silver part on the right hand side?
Cutting the pipe in half will allow the nuts to be removed and used again. The olives on the pipe are then useless.