Are you saying that the starter motor spins but the engine does not? If so, that’s a starter motor issue and it needs to be replaced.
If you mean the starter motor does not engage in PARK (I assume an auto trans) that could likely be a faulty neutral safety switch.
As to the running ragged on startup that sounds like a loss of residual fuel pressure. With fuel injected cars pressure must be maintained in the fuel lines and injectors so as to allow an instant start. If pressure is lost the result will be sputtering and coughing until the air is purged out of the lines and replaced by gasoline after a minute or so of running.
Think of a garden hose lying in the summer sun and water spitting out of it until the air is purged. Same thing.
The usual cause of fuel pressure loss is a faulty check valve in the fuel pump. This means a pump replacement. Other causes of pressure loss is a leaking fuel pressure regulator or a fuel injector sticking open.
Hope some of that helps you out.