The DI ticking from the injection system in my Z06 LT4 V8 is significantly louder when cold, the noise from the 3.6 in my ATS is slightly louder when cold. I suspect this difference across engines is related to clearances and the changing density of components between cold and warm leading to better sound conduction with a cold engine. I haven't looked at the tables but your 2.0T engine may call for slightly greater than normal rail pressure when at cold idle to provide for higher injection pressure and the richer mixture needed to quickly raise the front cats to operating temperature. Even modern synthetic oil is still slightly thicker at very cold starts and is better at conducting sound so it plays a role.
As long as you don't have two different rhythmic ticking noises, your engine sounds normal. All of the gas DI engines have this noise from the cam driven high pressure pump and the difference between engines depends upon layout, level of noise shielding, and to a small extent individual build variance.
A couple of other typical reasons for non-fuel system cold start ticking are the belt tensioner and in rare cases spark discharge from a damaged module or connecting wire. If it is due to spark, the problem should be noticeably worse in damp weather but this was typically a problem seen in older cars that don't use coil on plug style ignition systems and had long wires from the distributor to the various cylinders.
As to cold start, as soon as the engine stabilizes it is time to start driving it with a moderately light load on it avoiding heavy throttle or high RPM operation. Warmup via idling is one of the most destructive things you can do to an engine, cold operation doesn't make the engine happy and increasing the length of time in cold warmup is bad. Cold idling washes down cylinder walls and contaminates the crankcase due to incomplete combustion and you also have moisture in addition to fuel dilution during extended cold idling. Oil warms more slowly than coolant so avoid heavy load/high RPM until the coolant has come up to operating temperature for a few minutes and this is particularly true for engines which don't have an oil to coolant heat exchanger which also serves as an oil warmer instead of cooler during initial warmup.
Rodger