The "r" sound in RP and non-rhotic accents of English

Hello everyone.

May I ask you what happens with the "r" in the following words (in non-rhotic accents like RP): beware /b?'w??/, care /k??/, dare /d??/, there /ð??/, share /???/, compare /k?m'p??/, careful /'k??f?l/, sphere /sf??/, figure /'f?g?/, and so on? In all of these cases the “r” is neither at the end of the word nor before consonant (rules that many BrE teachers teach for silent "r") – still, it is silent. Are there any rules that can be applied in these cases? What about: very, necessary, arbitrary, and so on - here the "r" is pronounced, but, even though in the middle of the word, there's no consonant before it (other rule BrE teachers teach for non-silent "r")? What's the rule here? What about the words: order, separate and the like? In "order", for example, the "r" is before a consonant - still, it is silent. On the other hand, in "separate" the "r" is in middle position, but there's no consonant before it - still, it is pronounced and therefore non-silent. What I am trying to learn is whether (or not) there are 2 separate rules for the “r”: one telling me when the “r” must be pronounced and one telling me when the “r” is silent. Am I missing something here? Thank you!

As I have spoken with other BrE experts, I would also like to ask you if the following conclusions are accurate enough and could be considered a rule for the pronunciation of the "r" sound (in British RP and non-rhotic accents of English):

1. "r" is silent in the following words: car, star, sister, mother, word, person, bird (/k??/, /st??/, /'s?st?/, /'m?ð?/, /w??d/, /'p??sn/, /b??d/) because it is not followed by a vowel sound.
2. "r" is pronounced in the following words: read, write, red, Rome, grass, green, very, separate (/ri?d/, /ra?t/, /r?d/, /r??m/, /gr??s/, /gri?n/, /'v?ri/, /'sep?r?t/) and also in berry, carry, arrange ('/b?ri/, /'kæri/, /?'re?n?/) because it is followed by a vowel sound.

Or, to sum up: /r/ (the phoneme, i.e. the sound as in red) occurs only before a vowel phoneme (in British RP and non-rhotic accents of English). In every other case, it is silent.

Thank you very much!