Apparently there are 2 common articulations of /r/ in American English, one retroflex, and the other dorsal. For me (AmE speaker), /r/ is dorsal, not coronal, so there's no such assimilation. This phone is called the molar or bunched r. It can be described roughly as a back-palatal or pre-velar approximant that's somewhat bunched up along the left-right axis. Approximate transcription: [ˈnoʉ̯˞θ]. John Laver transcribed this sound using the symbol [ψ].
For speakers that use the retroflex articulation, it's possible to slide from a retroflex [ɻ] through [ɹ] to [θ] in one smooth motion, so I don't expect any special assimilation of /r/ other than this.
I do assimilate /n/ to interdental when it immediately precedes /θ/ or /ð/. However, the /n/ in north is separated from the /θ/ by two segments, during both of which the tongue's corona retracts from the alveolar ridge; there's no reason at all for it to assimilate. (During articulation of [oψ] I think the tip of my tongue is below the alveolar ridge away from the teeth, roughly at the level of the lower teeth.) I see no difference between either the /n/ or /r/ in north vs. Norse.
About the pronunciation of /ɔ(w)/ in north, or in gnome vs. norm: I exhibit the same monophthongization of /ɔw/ to [o] as JSBձոգչ noted. Alternatively you can think of the /r/ as a semivowel that replaces the /w/-offglide of /ɔw/. Also the initial vowel in the resulting diphthong [oψ] (as in norm [noψm]) is distinctly raised, not lowered, and also more rounded, relative to /ɔw/ (as in gnome [nɔʊm]). In Arthur, the vowel /a/ doesn't seem to be changed by the following /r/.