I have seen a lot of Klingon ships lately in Fed space (Like around DS9)

Are these Federation Klingon Characters in Klingon Ships?
So Federation characters can get Carriers if they are Klingon characters?
Can Federation Klingon characters do the Federation Area Missions?

DS9 is not exactly "Federation Space" (it's not blue)

So no, those are Klingon ships (they have the "IKS" prefix and no Federation Klingon characters are restricted to Federation ships.

Missions that were part of the "faction agnostic content", meaning "Featured Episodes" can also be played by the Klingons (and their allies) but Federation-exlusive missions remain exactly that: Federation exclusive.

Here's a list of missions the Klingons have:

http://www.stowiki.org/List_of_missi...ingon_Missions

No.
They seriously expanded upon the sectors Klingons can fly around in.
Basically, the only sector off limits to them is the Sirius Sector.
Feds still can't get into Omega Leonis.
Naturally, if a character comes along that has Diplomatic Immunity, either because they earned it or it has been temporarily granted to them, they can get into these normally off limits sectors.

We've had that ability for ages.

But OP, KDF players have been allowed to go practically everywhere for months now, the now old Red Alerts opened up the rest of the map to KDF players.

*watches Jem'pok and Quinn, prancing across the meadow, hand in hand, music playing*