How is the pronunciation of r before th? Specific case: "north"

Let me take your conjectures one at a time:

• a or o are lowered in the mouth so that we have /ɑ/ or /ɔ/.

This may be true, but I suspect that it's subject to quite a lot of dialectal variation and varies wildly depending on the other surrounding segments. All English dialects neutralize some contrasts before /r/, but the exact number and nature of the mergers vary a lot.

In my idiolect, the primary difference in articulation before /r/ is actually monophthongization. Taking the gnome/norm minimal pair, gnome has a diphthong which is approximately [əʊ], while norm has a monophthongal [o:].

• the mouth becomes less tensed

I'm not sure what you mean by this. Less tense compared to what?

• the r is very loose

"Loose" is too vague for me to comment on. The /r/ clearly does not become dental. In rapid speech, the /r/ may become very brief, to the point that its presence is largely marked by the preceding vowel allophone.

• n in north becomes dental!

This definitely does not happen. Dental/alveolar place agreement never intervenes across vowels.

The point of constriction in retroflex consonants per se tends to be alveolar/postalveolar. I'm afraid I don't know of any actual data off hand, but I dare say it would not be surprising to find that the point of articulation of /r/ is very slightly fronted before dental consonants (but still within the alveolar region).

On the other hand, while I guess physiologically possible, I think it would be hugely unusual for the articulation of a retroflex to be so far fronted as to warrant being called "dental". On the other hand, I believe some languages such as Hindi-Urdu have retroflex and dental /r/ sounds contrasting with one antoher.

(And I would be very keen to stand corrected if anybody knows otherwise.)

P.S. You might want to listen to how speakers pronounce "for the" in the sample recordings in the Speech Accent Archive (the link is to one speaker, but several have been recorded).

In my dialect (fairly generic AmE), your premise is not true. There is no difference in the pronunciation or the articulation of the 'n' between "tense" and "tenth".

Likewise, there is no difference in the pronunciation or articulation of the 'r' between "normal" and "north".

I wonder if that is not so much an "r" + "th" phenomenon; might it rather be a postvocalic r drop?

You probably are familiar with William Labov's study of social classes and the postvocalic r, but I will mention it for the general audience.