Irving gave you a great list of things to look for. I'll add a few.

1. Make sure the contractor pulls a permit and that you are present and questioning the inspector from the building department.
2. Ask for a copy of the contractors certificate of insurance before they do any work or you pay any money. Might even be worth calling the insurance company and verify the policy is in force. (I would). The last three roof inspections I have done for clients had some issues with the contractor, so better safe than sorry.
3. The issue of adding insulation is connected the the roof ventilation. If you have vents installed in the eaves/soffits, the extensions need to come above the level of insulation, so adding more after the fact, you don't want to then defeat those new vents. Even in winter, the airflow will be out of the ridge vents and snow infiltration is not an issue, provided they are properly installed and balanced with the soffits.

If you are going to add insulation and vents, a powered vent in the center of the roof structure with a humidistat and thermostat might be something to consider. They can save you a considerable amount on your summer cooling and will essentially become passive in the winter.