Is it explained how Starfleet found Khan or did they just happen across him randomly?
Yes, although I wish they had explained it slightly different, through the Kelvin's destruction.
INTO DARKNESS takes place one year after STAR TREK. So, within one year they found Khan, revived him, introduced him to the technology of the 23rd century, let him develop all kind of super-secret programs and build the Vengance within what... less than a few months?
Is it stated specifically that it was within that timeframe?