Unknown PIE root
Do Dutch and German realize "r" dorsally because of French/Frankish influence? Proto-Germanic *r was similar to Proto-Indo-European *r, right?
Yes, it was. Dutch r is pronounced differently depending on speaker and region. There are quite a few varieties. The standard pronunciation is the historical one, as an alveolar trill. But there is also the uvular trill, uvular fricative, alveolar approximant (as in English), or a central vowel resembling a schwa or a more open schwa (as in German).
I wonder whence the alveolar approximant derives. It's odd how Dutch and English are virtually the principal languages which feature it. It's also odd how here in America the respective vowels are rhotacized. The vowel you describe in German (I think IPA(key): [ɐ]) and the alveolar approximant in Dutch are what I thought were the realizations of /r/ at the end of the syllable (e.g. the "er" /(ə)ɹ/ in neder).
It's not that hard if you think about it. The alveolar trill has the same place of articulation. So if you articulate it a bit weaker by not letting the tongue fully close the air stream, you have an approximant. I think some dialects of Norwegian and Swedish also have approximants, and that kind of ties in with the retroflex assimilation that is found in many of those areas. If you imagine r and s falling together into a retroflex, there is almost automatically an approximant before the retroflex consonant.
It's actually easier conceptually for a trill to become an approximant, than it is for an alveolar consonant to become uvular. I'm not really sure where the uvular pronunciation comes from but there is a whole area in Europe that uses it, including areas of German, France, Danish and Swedish too.
That makes sense. I suppose the area using uvular trills got the phone from a common source. I've always assumed that Danish got it from German and its close relatives, and that languages like Pre-French used alveolar trills. Could it have possibly been from a non-Indo-European substratum (although it's a long shot)?