Question
Is the root system of a sweet gum tree deep or shallow?
Is it true that large sweet gum trees rarely fall over, and mainly lose branches when the tree is compromised?

Answer
I was taught in school that the American sweetgum (Liquidambar styraciflua) has a shallow, extensive -some would call them aggressive- root system.  Imagine it is like a wine glass on a dinner plate, the plate being the root system.  Other shallow-rooted trees found and planted extensively in the US are the elm, beech, linden, maple, and honey locust.  

As to it falling over, they can get up to 75 feet tall under the right conditions, with a spread of 40 to 50 feet.  The shallow, expansive root system is an advantage in holding on to the soil with that sized tree.  Sweetgums are susceptible to few pests, and tolerate nasty soils, except alkaline.  All trees will fall over, given the right conditions. It is not particulary noted for this, however.

Branch drop is a concern with fast-growing trees that have been allowed to grow into poor form. I recently lost a maple to wind damage.  There is a condition called "summer (or sudden) branch drop" where a drought-stressed tree will shed large limbs, but the sweetgum tree is not ranked as susceptible, on the list that I have.

Hope it helps.  If you still have concerns about a specific tree near you, consult an ISA-certified arborist in your area. ~Marc