Minecraft – Would you be able to add an NBT tag to an already ‘spawned’ mob

What you want to do is impossible. Placed banners are blocks (well, tile-entities), and blocks cannot have names that appear above them, only entities.

A workaround perhaps could be to summon an invisible ArmorStand or WitherSkull with the name you want, and have that on the banner so that it appears as if the banner has a name. Try this command:

/summon ArmorStand ~ ~ ~ {CustomName:"Banner",Invisible:1b,CustomNameVisible:1b}

The minecraft wiki has pages on NBT data, including that of tile entities. Rather than guessing, it's a good guide to check what tags an entity/tile-entity has.

Entities and Tile-Entities: http://minecraft.gamepedia.com/Chunk_format

Players and Items: http://minecraft.gamepedia.com/Player.dat_Format

This answer was written prior to the release of 1.13 and "The Flattening" that came with it. An updated answer can be found here.

You're actually pretty close, and definitely on the right track. Instead of {Inventory:[{id:<block ID>,data:<data value>}]}, use {Inventory:[{id:<block ID>,Damage:<data value>s}]}. The s after the data value is important since it's stored as a short, not as an int. (As an aside, if it were stored as a byte, you would append a b instead.)