Most Emblem and calling card (callsign) unlocks happen by doing certain tasks within the game. If you are looking for specific multiplayer emblems, any of the additional special emblems are unlocked through black market purchases from cryptokeys (obtained by playing multiplayer matches).
If you are looking to create emblems, I recommend checking this site out as it houses quite a few tutorials on making emblems. http://www.codplayercards.com/
As for the clan tag, it will typically unlock at a certain level, which you are a higher level than I am, and I have it unlocked, so I would assume it should be unlocked for you. One idea is to make sure that you have "User generated content" turned on.
On the main multiplayer screen, click on the menu icon in the bottom right. Then go to the options tab and go to the content filter option. Set "User generated content" to enabled and this should allow you to update your clan tag (and may resolve issues with emblems for you too). See this youtube video if my directions did not make sense.