If I have the Settlers of Catan 5-6 player extension (for the base game), does that mean I can use the Seafarers or Cities and Knights expansions with 5-6 players?
Seafarers and Cities & Knights have their own 5-6 Player expansions in addition to the base game 5-6 player expansion. Think of them as expansions to expansions. So to play a 6 player game of Seafarers you will need the following:
Base Catan Game + 5-6 Person Base Expansion + 4 player Seafarers Expansion + 5-6 Person Seafarers Expansion
No--not easily at least. Each expansion has it's own 5-6 Player Extension, which you would need in addition to the base 5-6 Player extension. (BTW, they use the term expansion for things like Cities & Knights or Seafarers that change the rules significantly, and the term extension for sets that just add support for additional players.)
Seafarers adds boat pieces for each of the 4 base-set colors, but does not include boat pieces for the 2 additional colors. These you could proxy, or you could buy the Seafarers 5-6 Player Expansion.
Cities & Knights has more player-specific equipment: knights tokens, city walls, flipbooks for each player... playing without the C&K 5-6 Player extension would be very difficult. Traders & Barbarians is even more extreme than Cities & Knights.