There isn't anything hanging down under the car that can clog. The air intake is high up on cmax. If the engine was running for 13 hours, the engine bay would be plenty warm enough to melt ice away. Most likely cause of the exhaust noise us the exhaust section joints have come away slightly due to the exhaust dragging on snow.
Ad to dodgy gear shift, that's normal in the cold, the oil in the gearbox gets much thicker when it gets cold, it'll free up after a while driving.
Best solution I've found for cold starts is to start the car and leave it running 5 mins before you leave. I use spare key to lock the doors and wait inside. Leave blowers off. Then get in and the engine will be warm, turn on heater blowers for instant hot air. Off you go.
Ford test in much colder climates than we get and sell similar cars on a lot more hostile places. It'll be fine!
2005 Focus C-Max 1.8 Petrol Ghia + Sony 6 disc, 2007 tail lights, Led bulbs all over and HID front fogs.