"Tugann na póilíní tuairisc don bhreitheamh."
Translation:The police give a report to the judge.
I'm going to assume that "póilíní" refers to police outside Ireland, seeing as we have "gardaí"?
For the most part, yes. A few exceptions would be Póilíní an Aerfoirt (Airport Police), Póilíní Airm (Military Police), and police in Northern Ireland.
Probably - it's a common enough feature of Connacht Irish
Gramadach na Gaeilge notes that "In Connacht, do is pronounced [g@], as if it were written go" (http://www.nualeargais.ie/gnag/do.htm).
You can also hear the same change in "ag dul", pronounced "agul" - https://www.duolingo.com/comment/11806896
Is there a good way to remember when there is a séimhiú and when an urú after a preposition plus alt, e.g. ar an, don, sa etc?
In the Caighdeán, those prepositions that are able to form prepositional pronouns can either eclipse or lenite after an (or after a combined form with an) except for idir, ionsar, and those that are mainly literary or archaic, such as seach.
What is the difference between bhreithimh and bhreitheamh? They both translate as judge.
The difference is that breitheamh is in the common case, and breithimh is in the genitive case.
In modern, standard Irish the common case encompasses the nominative, accusative and dative cases, because there is no difference between the forms of nouns in those three cases.
don bhreitheamh is correct and don mbreitheamh is incorrect (though it is used in some parts of Kerry).
Eclipsis does occur after some most) prepositions and an, but do isn't one of those prepositions.