Firstly, welcome to the forum.
I had exactly the same problem with the CD player as you - it would read and play mastered discs, but hardly any of my CD-Rs. Then, it stopped playing anything. I fitted an aftermarket system (Pioneer) and have not look back since.
The point of saying all this is that the Trip Computer and Check control system work fine without the original system being connected. However, you will lose the RDS clock facility (which I found quite good actually) and the display will no longer show the radio station or disc / tape readout. This is replaced by the date. I am not too bothered by this although you can purchase an adapter to drive the display again. They come up on e-Bay from time to time and are made by Connects2. This is not as easy as it seems though; 1. the are quite expensive (over £50 I think) and, more importantly, 2: there is hardly any room behind the head unit for additional cabling and interface boxes as you will discover. For these reasons, I left it alone. You will also loose the steering wheel buttons, and yes you can get an adapter for these, but again I left them disconnected.
There are some Car Audio experts on here, Laidback and Hilper being two that come to mind.
Hope this helps