The reason you can’t start the engine in second gear is for safety reasons. Car makers fit what’s called a Neutral or Park switch, which means the engine won’t crank unless the car is in Neutral or Park. That prevents the car starting in gear and taking off.

If the lever’s connection to the transmission has failed or become excessively worn, you can get the symptoms you have because the lever won’t select Neutral or Park. And therefore the engine won’t start.

However, it’s highly questionable that this is connected to the limp-home mode. That’s usually caused by a problem with the driveline that threatens to do more damage if you continue driving. The gear lever problem sounds more like a simple mechanical issue. Did both things happen at the same time? If not, they’re very probably not related.

Those codes are reasonably common ones. The first two (0193 and 0089) generally relate to a problem with the fuel rail, specifically the pressure in that rail. This is obviously critical to how the engine runs.

The third code (0401) is more commonly associated with the flow of the EGR valve. If this valve is damaged or blocked, then this fault code would likely show up. As such, you may find you have two separate problems, rather than a single one.

Low pressure in the fuel rail would certainly explain why the performance is off, and the overheating could easily be associated with an EGR valve failure. A blocked EGR can raise combustion chamber temperatures while a failed EGR can actually leak coolant and allow the engine to overheat that way.

If the problem goes away when you restart the car, it’s reasonable to assume that there’s something electronic that’s responding to the rest, or there’s something like a crank angle sensor that is getting hot, delivering bad information to the computer and then working fine again once it’s cooled down.

Have you had the vehicle scanned for fault codes? This should probably be your next step rather than simply changing more components that don’t turn out to be the problem. Meantime, don’t rule out things like the spark-plug leads or even a failing fuel pump. How about something from left-field. I’ve personally seen a dirty fuel filter allow a car to run perfectly for the first 10 kilometres, until the gunk built up on the filter element and stopped the fuel flow. Switching the engine off and then back on again, allowed all the rubbish to fall away from the element and the car would be perfect for another 10km. Until it wasn’t.