The bill of Rights is part of the US constitution, (the first 10 Amendments), so they will be interpreted by the same institution as the one that interprets the constitution generally.
This would be the Supreme Court of the United States - specifically, the Supreme court checks whether the laws passed by the congress are compatible with the constitution
In general yes, the states honored their agreement since the Articles of Confederation demanded practically nothing from the states--meaning that it wasn't hard for the states to uphold their end of the bargain.
