How old is Stack Overflow?
The private beta for SO launched July 31st-ish 2008. The public beta started September 15th, 2008.
What did the first version of Stack Overflow look like and how much has it evolved since?
Aesthetically it hasn't really changed. Visually very small, incremental enhancements have been added periodically. Most of the work happens at the functionality level.
How many days/months did it take to build the first Stack Overflow?
(That's a joke, AFAIK it started ~April/May and the beta launched end of July, so pre-beta work was ~16 weeks, and continuous additions and refinements since then).
Found under the entry "Stack Overflow Beta Design" is an early (pre-beta) image of the design.