I live in the northeast of England where it can get quite cold. The bonsai has been outside until a few weeks ago but I thought if the frost got to it, it would damage the roots and kill them. It is only in a bonsai pot not a large garden container.
Despite being sub-tropical Chinese Elms are very hardy although this can vary from tree to tree. Mine have frozen and come through our winter fine. If you are concerned about this another option is to keep it in an unheated garage or shed of some sort. That is how I am handling mine in recent years.You mentioned a cold porch, this sounds promising.
How do I tell if there is a problem with the soil
Good bonsai soil is very gritty and free draining. To a novice it might appear to be little more than gravel, with perhaps a bit of bark.
my son owns a bonsai too and he keeps telling me he finds 'worms' in the soil so he keeps repotting it.
Worms could mean quite a few different things. How is the soil becoming repeatedly infested? Is the pot put on the ground perhaps or is your son using soil from nature that may already be infested. If he keeps re-potting it with too great a frequency eventually it will become weakened and fail. The cure may end up being worse than the disease. I suggest that he identify the culprit and, perhaps more importantly, the source of the reoccurring problem.
I've suggested he uses 'provado' systemic insecticide but it doesn't seem to help. Any ideas?
If the insects live in the soil a systemic may not be the best choice. Again, a proper identification will help greatly in choosing the correct product.