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You can ask them to report your mortgage, but I don't think they are under any obligation to do so. There are costs to reporting information. Lenders choose whether or not they want to incur those costs. I don't know if there are any special rules requiring mortgage lenders to report.
As far as the CC is concerned, I would think having a mortgage that you pay responsibly would be on balance better for you. If you're looking for a high initial limit, credit unions are known for giving higher initial limits than banks.
Walt is correct, they are not required to report your mortgage payment history, but if they do then it must be accurate. I am not sure what is required on the credit card application, but most ask what your current housing payment is so your mortgage payment amount would be known either way. You probably want to ask this in the credit card section since it's a credit card qualifying question.