The problem with Latin grammatical cases is that, they are generally ambiguous between different cases. plural nominative masculine is the same with singular genitive masculine for example. Or singular dative feminine is the same with plural nominative feminine and singular genitive feminine.
So the ambiguity makes it complex, very hard to learn for rest of ...
It looks like a comparative (cf. facilius, melius, and many others) but it is in fact a genitive.
Thus unius libri is "of one book".
The word unus has an unusual declension:
nom: unus, una, unum
acc: unum, unam, unum
abl: uno, una, uno
The same genitive in -ius is used by a couple of pronouns.
There is no comparative form of ...
This means "Ours by the sea"* (or, if you want to supply a noun where Latin will let an adjective do the job of a noun "Our place by the sea"). Thus, nostrum is nominative and mare ablative. It would be a nice motto for a family with a sea-side home, or for a sea-side city.
Alternative interpretations, with nostrum modifying mare, are ...
Another possible interpretation would be that pro (alternatively written proh) is an interjection of sorrow or desperation, meaning something like “alas,” “alack”. While often used with a nominative or vocative (pro dii immortales, pro sancte Juppiter etc), it is also found with the accusative: pro deorum hominumque fidem! (which makes sense because the ...
The entry for pro in Lewis & Short mentions at II that the preposition pro comes with the ablative but remarks that accusative is possible in late Latin.
As you quote a coat of arms, influences of late Latin are certainly a possibility.
I don't know what the relative frequency of the two cases with pro is in any given era — apart from the accusative ...
It's the accusative of time, answering the question “how long?” (And not “when?” or “during which time?” – that would call for the ablative.)
“I thought myself the happiest of people for so many years.”
One might consider the ablative defensible here, which would then answer the question “within which time,” although that would strike me as an unusal choice, ...