In standard Irish (and Connacht and Munster), this should be B'ealaíontór é an fear sin. It's only allowed to drop the é in Ulster Irish. It's the first example under 'The Classifactorial Clause' here
Strangely, B'ealaíontóir é an fear sin is the translation in the reverse exercise with B'ealaíontóir an fear sin given as another correct solution.
1) Is the first word a contraction of "ba" and "ealaíontóir", or is it "bhí"?
2) What is the difference between "ba" and "bhí"?
It is a contraction if "ba". "Bhí" is the past tense of "bí" (tá) meaning "was". "Ba" is the past tense of the copula, "is" and means "was" or "would be"
The “would be” meaning of ba is its conditional mood rather than its past tense.
Do sin, seo, siud always go at the end because they're direct objects?
sin, seo and siúd aren't direct objects.
tá an fear sin ag léamh
an seachtain seo caite
tá siad siúd imithe le fada
Does Irish have the idiom "used to be"?
Unlike English, Irish has an actual past habitual tense.
Cheannaítí pasta san ollmhargadh
Ghoidtí na leabhair gach oíche
Bhínn sa leabharlann gach oíche