What (if any) is the relationship between your Active and In Decline race?
No, powers cannot be shared between races.  They are specific to one race only.

Sometimes though, powers carry on in to decline.  When you flip over your race/power tokens to go in to decline, it usually has symbols or graphics on the back that show what powers remain in decline.  If it is just blank then you have no decline powers (See backside of Dwarves, Ghouls, and Trolls race cards here (Courtesy of BoardGameGeek.com).  See also spirit power).  However, these decline advantages are still specific to that particular race.

For the second question, how does an active race occupy a region owned by your race in decline?.. 
Page 4, section 2, 2nd Column, says:


  A player may choose to conquer a Region occupied by
  his own In Decline token, if he wishes to: He will lose that
  token, but might gain access to a Region that is more
  profitable for his new Active Race tokens to occupy.


This is the only for your active race to occupy a region in control of your decline race.  The same rules apply as conquering any other region.