I am level 12 which means I can start doing some court politicking at the Winter Palace masquerade. I know this progresses the main story line but will doing this part prevent me from completing any other quests at the War Table? For example, I haven't finished all the stuff with the Gray Wardens yet. Will I still be able to do it after completing this quest?
No quests will be blocked at any time. The only exception to this are the romance options, which become blocked after completing the main story.
If anything, I'm impressed at DAI's ability to adapt to the player doing an unexpected order of quests. I ended up clearing Crestwood before even entering the Fallow Mire. When introducing the risen undead in the Fallow Mire, scout Harding specifically added "but you should know how to deal with them due to your experience in Crestwood".
There have been other instances where the game subtly acknowledged my past actions, even when the two events were expected to happen the other way around.