That is really good news that you have confirmation now of the divorce. Something at least that you can now work from. With the reference numbers etc you have been given, I would think you should be able to perhaps order a copy of the paperwork from Edinburgh. Did you check on this at all? If the divorce reference show on the the National Records of Scotland (from the site linked earlier), you can simply send in an email in to NRS to request a quote for copies of the file. In this case, I couldn't easily see a ref to it just working from the surnames we have, but maybe you can make enquiries for this.
I would imagine that the divorce papers may hold more details regarding the adultery (name of other party?) and perhaps details of the child's birth. Hard to say always with all of these documents until you view them really.
Also, hard to say what surname the baby would have been registered under too. It could have been under Gray, Connel(l) or the reputed father's surname. For the child to have been registered under the reputed father's surname, he would have had to have attended the Registrar's office to confirm he was happy for his name to show on the child's birth certificate. If he was not there, mother could not have added his details.
In Scotland, married women (even today) never loose the legal right to use their maiden surname. This is the reason that maiden surnames are so prominent in Scotland. You can search for the death of a married woman on Scotlands People, for example, searching with both surnames. Many documents will show a woman with both surnames, eg. Janet Connell or Gray would be the norm.
Not helping necessarily here though in your searches
All too recent due to privacy and data protection to be able to search in detail online.
Monica
