Unless there is a reason for the gas company not to do so, I would have them do the underground run to where your existing black steel pipe is and have your meter there, rather than just going to the closest corner. It might cost a little more and be more work digging the ditch for the plastic pipe they use, but I think it would be worth it to have it done right, the first time.
You might want to do some thinking about possible future modifications you might do to your house and yard, and plan the underground gas line route to avoid those future plans. I wish that when we put a natural gas line and electrical lines to my new house, that we had put them on the other side of the foundation. The way the buried lines are now, I cannot add on to my garage, since I would be covering the buried lines. I wish I had thought of that before I did it!
And everything that works with propane now will have to be modified or replaced to work with natural gas. It takes a LOT more volume of natural gas to produce the same amount of heat as propane.
A friend of mine was fixing up an older mobile home for his widowed mother to live in some years ago. It had gas appliances and they hooked it up to a fairly large propane tank. The first time they tried to light the cooktop in the kitchen, they had flames nearly to the ceiling and darn near burned the thing down. That was when they figured out that the gas appliances were set up to use natural gas. It took a service call from someone who really knew what he was doing working with gas to get all the appliances working properly with propane. The service call was not that expensive and neither were the needed parts.
You also probably will need to have the modified system inspected, so it might pay to find out what the inspector will require beforehand. I think you will enjoy being hooked up to natural gas--we sure do. Lots easier and lots less expensive. Good luck!