Can I charge elderly mother rent and sitter services by family member

You can charge her (perhaps offsetting the costs for the care she provided during your infancy and childhood).  You should have a family caregiver agreement specifying time and task, giving each of you an "out," keep careful records and deduct for Social Security, Medicare and payroll withholding (There are services which will do this for you.)  You should make sure that this is done in a way which will not affect VA and Medicaid benefits she may need now or in the future.