In the word “never,” how is the final “r” pronounced in most American accents?
It is clearly pronounced /ɚ/. -> in UK is silent.
In the word “never,” how is the final “r” pronounced in British Received Pronunciation (RP)?
It is silent unless followed by a vowel sound.
In the word “rain.” Is the “r” pronounced in both American and British English?
Yes, at the beginning of a word it is pronounced in both.