You can actually hire a car when you are on an employment visa. Most of my collegues have done it! You only need a copy of your passport and license. However, as you have said, you will need to convert your license to a local one once your residence visa is issued.
I cannot comment on your encounter with Immigration. My company used Marhaba and they forgot my employment visa at the desk. They had to go and get it as soon as I said I was here for employment! We went back to the desk after that and the immigration officer clearly said that I had to do an eye scan. The employment visa was then stamped by the guy who did the eye scan (and also rudely told me to get out of his chair as the eye scan was complete!) and again by the immigration officer, who as you say, put a stamp in my passport and then sribbled in arabic next to it.