"Can I expect a synthetic engine oil to extend the service life above that of a mineral oil? Also, can the service life of oil be extended by increasing sump capacity? For example, if we double the sump capacity, can we then expect the service interval to be doubled or is there a ratio?"
Synthetics can have extended drain intervals. However, the factor at which the extended drain would be assumed depends on many factors, including contamination control and exclusion, typical operating conditions of the vehicle, temperature, etc.
A larger sump equates to more additives and detergents for the same engine. Therefore, a larger sump will extend the service life. Assuming all conditions are the same, twice the oil in the sump should provide twice the service life.