Basement egress through office
Yes, this allowed according to recent versions of the IRC building code. (If your state or region has a different building code you will need to look it up yourself or specify.)

To quote the code:


  R310.1 Emergency escape and rescue required. Basements and every sleeping room shall have at least one operable emergency and rescue opening. Such opening shall open directly into a public street, public alley, yard or court. 


So in short it only counts as an emergency exit if it leads directly outside. It can't just be a route outside.

Since there are no bedrooms in this basement, a single exit from the basement is allowed. If you convert the "office" to a bedroom, that is also allowed, but you could not convert the "family area" into a basement without adding an additional exit. (You may wish to read the rest of that section since there are additional requirements and definitions, but that is the gist of it.)

However the short answer to any code question is always "whatever your local inspector says", since they are the ones that have the final say or not.