I'm not sure about VA law but in my opinion I think the contempt issues are separate. If you believe your ex to be in contempt you need to file that motion and let the judge decide. If found guilty I think your chances of modifying custody will increase. However, I also believe these issues to be dealt with in court need to be established prior to the hearing; I'm not sure that you can bring in a whole laundry list of issues without prior notice.
In regards to your first two issues, the judge would probably prefer that the two of you attempt to create an alternate schedule to accomodate Dad's schedule, as that would be what is best for the child. If you can show that you both agree on changing it up the judge is more likely to approve your request.
Your third issue will be hard to pursue, unless the child's grades are slipping or they are facing school or legal issues, just stating "discipline problems" won't go too far. You'll often get a "kids will be kids" or "it's due to their age" type of answer.
The 4th issue should have been brought up before the original order was approved. Did you have a chance to bring this up in court or did you not know about it until later? I think this and issue #5 are also separate issues which will have to be addressed via a contempt motion.
I obviously can't give you any legal advice but this is my opinion based on my husband's case in CA. The judge will always consider best interest of the child first and will want to see concrete evidence for the issues you bring forth. Do you keep a journal or log of each time he violates the order? How can you prove he is dropping the child off with a babysitter? These are things to consider before bringing them up. Good luck.