Does the pronounciation of post vocalic 'r' in english is linked to its position in the word?

^^^ what he has said.

British person would say 'row' about the same as American,

while word 'third' would englishman pronounce "t-hhhd"

Not in American English, where the position of the r doesn't matter. In British English, it is pronounced if it precedes a vowel, but not otherwise.