> Assume for a moment that the select ID from TABLE_A returns 10 rows. If the client simply calls OUTER_STORED_PROC but never reads any of the result set, does INTERAL_STORED_PROC get called only once, all ten times, or never?

It is very hard to predict. The SP will be called any number of times
between 1 and 10. Exact number depends on many things, including server
type.

SY, SD.