Earlier quoted context omitted.
For "rural", I'd expect there to be significant differences between rhoticizing and other dialects.
I don't think so, as both "r"s are sounded (like at the start of a word in a non-rhotic accent). It's when the "r" is at the end of a sound that there's a difference: "are", "our", "letter", "arctic", "architecture", etc. (none of those "r"s are sounded in my non-rhotic New Zealand accent, edit: unless followed by a vowel sound, like "letter and").
Other Southerners with thicker accents would run it together like /roo'-əl/: two syllaables, schwa in the 2nd, no perceivable 2nd /r/.
There may even be a 3rd way: /rulr/ rhymes with earl.