So assuming I have a trainer that is not lazy, will temperatures in the single digits or at 0 affect them jogging horses or doing training miles?
You have a great question---- Some days the weather, be it low temps, to much snow, slippery surfaces, or high winds.... a good trainer will way the risks to the horse vs the possible benefits and smart trainers make the proper calls in regards to those days off.....

The money hungry trainers that insist on training 2 year olds to meet stake deadlines are part of the overall problems in this business when it comes to the well being and longevity to the health of the animal.... One should never race a horse till they are at least 3 years of age... IMO! If you think you have a good 2 year old save them till they are 3 yrs old---they will be that much better and stay sounder and healthier as well!!