The DA42 will climb quite well on one engine at MTOW and is pretty docile as mentioned. If you trim out the rudder it really isn't much of an issue flying on one engine especially once you have feathered the prop (master switch OFF). However like all twins, if not handled correctly then things can go wrong pretty damn fast.
There was an incident some time ago where a DA42 took off after having a flat battery. They started it with ground power then took off. When the gear was raised the ECUs both tripped out (lack of power) and both engines stopped. However the procedures were changed so that if you have a flat battery then you only start ONE engine from ground power and start the other normally. This ensures there is enough power in the battery to raise the gear without causing low voltage conditions.
I haven't heard of many mechanical problems to the Thielert which have caused a catastrophic failure as in the Lycoming type of failure. I did once have a turbo boost problem on one engine when climbing through 8000', the right engine power reduced to 75-80% and kept reducing with altitude (left stayed at 100%), but it wasn't an issue low down.