WichoP South Texas
Unfortunately, your photos did not come through. I will be unable to Identify the damage.
This can be due to several reasons. Low humidity, dry soils, and wind can cause this. Other things such as bacterial, fungal, and other infections can cause this, as well as nutrient deficiencies.
Here is an article that will help you to know what kind of care they require:
https://www.gardeningknowhow.com/houseplants/ti-plant/growing-ti-plants-outdoors.htm
Is it best to let it start in water?
Starting them in water is the most common method. If your log root can be fitted with the proper soil for the plant with a pH range between 5.6 to 6.5, and if the plant has good drainage it would probably work. I would expect the root to provide everything the plant needs, but to serve as a vessel for a well-nourished plant with the proper amount of light, I don't see a problem.
https://www.gardeningknowhow.com/houseplants/ti-plant/growing-hawaiian-ti-plant-indoors.htm
Will I have to dig up the root all the way or is there an other way of killing the plant without killing the grass.
I would try a selective herbicide. It doesn't kill grass but it may kill your plants even though they technically aren't weeds.
Im trying to grow a ti log but im unsure whether im during it correctly,how can I tell if it is sprouting
You will see sprouts, and new leaves forming.
https://www.gardeningknowhow.com/houseplants/ti-plant/growing-hawaiian-ti-plant-indoors.htm
My outdoor Hawaiian Ti plants got a slight freeze and look like they need help. Should I cut them back?
Do not cut them back till all the inclement weather is past. Here is how to care for cold-damaged plants:
https://www.gardeningknowhow.com/plant-problems/environmental/how-to-treat-frozen-plants.htm
My grandson grabbed my ti plant and broke the stem and remaining leaves off of the plant. Can I save it?
It has a chance to sprout new leaves. It would have a better chance if there were a few leaves left, but just continue to care for it normally and hopefully it will respout.
The stalk of my Hawaiian Ti plant has mold or green fungus on the bottom of the stalk.
Unfortunately, your photos did not come through. I am unable to see the damage.
Some green fungi, like Trichoderma species, are beneficial to plants, where as some species of Penicillium can be detrimental to plants. I can tell the difference by smell. One smells like detritus, where one smells like citrus peels, but this is not going to help you much.
Other alternatives are lichens, mosses, algaes, bacteria, and other, actual, pathogens.
Algaes will gunk up your soil and prevent water and nutrient uptake. Bacteria will usually either destroy your plant and soil, or be harmless.
If you have a microscope handy, this is how I tell the difference between certain microbes in the garden. That and a few of the most common stains will give you close enough of a ID to know whether you have a problem or not.
Otherwise, you can take samples to your local extension service, and they may be able to help: