Are there any mechanical / safety issues arise when you do not put trims on car tires? Even when using the car in winter temperatures? (Say -15C to -20C)
No, hubcaps are purely cosmetic and do not need to be fitted. Some designs improve fuel economy and reduce wind noise by having a fairly flat surface but a car with one or more missing hubcaps is perfectly safe and in the UK, for example, hubcaps are not required to pass the annual MOT (safety inspection).
I have driven on steel wheels with no hub caps in the harsh Finnish road environment for about 6 years. I had hub caps on neither the winter wheels nor the summer wheels. No problems encountered. The reason for doing so was that I didn't want to remove and reinstall the hub caps two times per year at wheel changes.
The car was of course inspected annually. No one there complained about missing hub caps.
Now I have aluminium wheels. On my previous car, the winter alu wheels once were so tight due to expanding rust or aluminium oxide that removing them required a rubber mallet. Hasn't happened on my new car yet, though. This is less likely to happen with steel wheels, which is a major advantage of them.
The reason for hub caps is that they make steel wheels look like aluminium wheels, and hence more expensive. Just cosmetics.
One winter (1978) I drove my then 11 year old car over the Cascade mountain passes in Washington state many times and for some reason I didn't have hubcaps or wheel covers on the front wheels. Before the snow was gone, I had to replace the wheel bearings on the front and I believe the reason is that slush (maybe salt?)was able to penetrate the "spindle" cover. Maybe the "spindle" covers were loose (they are a force fit and had been removed several times for brake work), and maybe there was another reason for the wheel bearing failures, but I think hubcaps or wheel covers provide added protection to the "spindle" cap. (I don't know the actual name for what I am calling a "spindle" cap.)