1. These two are a couple. They are in a relationship. - depending on details (if not a married couple): Они встречаются. or Они живут вместе. Or in general: У них близкие отношения. other versions are possible.
2. These two were a couple (i.e. not married couple). They broke up. Они расстались. Они разошлись (the latter is also applicable to a divorce). Они разбежались (more slangy). Они порвали друг с другом. other versions are possible.
3. The reason why the broke up is because they were fighting all the time (i.e. they were arguing and shouting each other, but they were not hitting other) - Они расстались из-за того, что всё время ругались (or: ссорились).