First of all, I apologise for the terms I'll be using, English is not my mother language, I hope I don’t confuse you.
I am not a technician, but I believe that not starting at all could be caused by a battery pole not being connected properly. In case the positive is loose, you could get such a situation.
As for the steering wheel, a bad battery might affect the power steering system, but it may have been caused by other factors as well.
A bad battery shows other signs as well. When you start the engine, the ignition (?) sounds weaker and slower (the part that gets activated when you turn your key). You could also notice your windows going up/down with "difficulty". You may also see dimming in your lights when turning on the defogging of the rear windscreen. Or even failure to central locking and alarm being armed.
Your car being new (as well as it's battery being new) don't necessarily mean that your battery will be fine regardless of other factors. If the car does not move for a long period of time, or if it is left with the lights on while the engine is not running, or if you defog with the engine not running, then you could easily get a battery failure. Your technician will have more details about battery issues advices etc.
Shaking could be caused by a damaged alloy, in case you hit a road bump hard. It usually occurs at higher speeds though, say above 75mph.
Anyway, if it turns out to be just a battery thing, that's great, you can have the battery changed (hopefully covered by your guarantee) and no more worries
Let us know how things turn out.