Learn English – ‘deep’ in ‘Beauty is more than skin deep.’

While some of the definitions and idioms mentioned/linked are relatively close approximations, they all miss the specific idiom "in too deep", referenced here: (to) be in too deep

This generally references being involved in something so thoroughly that you become unable to extricate yourself (you can't escape); it is meant to allude to the experience of being in a mud bog or shallow quicksand (not enough to bury you, but too deep to escape), or to wander into a cave or woods so far that you no longer know how to return.

Note this is not precisely the same meaning as being "in deep water" or "being thrown into the deep end"; these latter two idioms indicate overwhelming challenge or great difficulty, or being forced to face a challenge beyond your experience or comfort.

Grammatically, you're correct, but the author here is using breathed deep in a poetic way. Some surrounding text is as follows:

For a moment Eddard Stark was filled with a terrible sense of foreboding. This was his place, here in the north. He looked at the stone figures all around them, breathed deep in the chill silence of the crypt. He could feel the eyes of the dead. They were all listening, he knew. And winter was coming.

This paragraph is describing the crypt, and the phrase "breathed deep" is describing the stone figures, not Eddard Stark's actions. This particular usage is very unusual even for a native speaker, and is done this way intentionally so, to add to the mental imagery that the author is attempting to convey. It helps to understand if you swap "breathed" for "buried", though it loses some of the unusual nuance.

As for using "deep" versus "deeply", that too is a creative choice. "Deeply" is what the reader (and typical grammar) would expect if "deeply" were in reference to Eddard Stark's actions, but by breaking out of expectations with use of "breathed" to account for their presence in the crypt, and following it with "deep", it serves to adjust the purpose of deep/deeply from their depth in the crypt, to how they were breathed into it.