I don't have children, but I teach children with Autism. It is very common for children on the Autism Spectrum to have symptoms of ADD, ADHD, anxiety disorders. Many doctors are hesitant to give a dual diagnosis. Kudos to your Doctor! Has your doctor given you referrals to the nearest Autism Society yet?
Do you have any specific questions or just looking for support right now?
If the parents of my students could give you advice, I bet it would be to find a family therapist (social, psycologist or psychiatrist) to help you and your spouse and other children deal with this. It's going to be a huge change in your life and you need to take care of yourself so you can best help your child.
Yes, it is common for a child to have traits of different disorders, or a main diagnosis with traits of other ones. It is harder to get a further diagnosis after the first main one, simply because further testing costs money and the attitude tends to be "there is no point doing further testing when we know that children with XXX usually have these types of difficulties". However lack of further testing to identify exactly what the root of any problem is, can mean that the child does not receive the right support or therapies from the right professional. For example a child that appears inattentive in class maybe like that because they have an auditory processing disorder, auditory processing delays, receptive language difficulties, sensory integration disorder, executive function disorder etc. Once you can pin point 'why' they are inattentive, you can support them. A child that has auditory processing problems may need lessons broken down into smaller chunks and for language to be kept short and simple and repeated to make sure he understands as well as using visual symbols and a visual daily timetable. If the inattentiveness is due to a sensory overload problem where the child has retreated into themselves because they cannot cope with the sensory demands of the classroom, then a low arousal environment, and smaller classes would help. If the inattentiveness is due to problems with executive functions and not language then the child needs supporting to help them plan, organise, and sequence their work. Does that make sense? In many cases, parents have to result to private reports to get a clear picture of what their childs needs are and how they should be supported and by whom, and to what level.
So having a dual diagnosis, and also evidence of any other difficulties will mean that your child should receive more support. But you may need evidence of that. And depending on how the education system works in your country, you need to make sure that he gets the right educational place that suits him.
It is also common for a diagnosis to change.