I think 'ar an spéir' is more usual.
I have never come across this expression; isn't 'tuar ceatha' more common?
Bogha sín’ (presumably a shortened form of bogha síne) is in Dinneen’s dictionary. The FGB has bogha báistí, bogha ceatha, bogha leatha, and bogha síne, as well as tuar ceatha and tuar báistí.
Ta coisul an bogha baisti sa dorcas! I think. I'm not good at this yet.
What are the words for plural rainbows?
bogha báistí is a genitive phrase -"a bow of rain". The plural is "bows of rain".