The reason that magic is detectable for underage wizards outside of school is because they have the trace on them, so they can tell when magic is being done. Many young wizards can get away with doing magic, because they live in a wizarding household and the MoM cannot tell the difference between who cast the spell without checking the wand (which not all magic requires). Once they turn 17 the trace is removed and so they can no longer track what that said wizard has been doing. The Floo-Network is entirely different to casting a spell, it is just magical transportation, which I believe they can choose and change which fireplaces are connected to one another through the MoM department, same as portkeys and disapparating.
The MoM knew that the dark mark had been used, because half of the ministry of magic was at the campsite that night for patrolling the hundreds of thousands of witches and wizards that has turned up to see the match. The ones that apparated together around Harry Ron and Hermione, were already at the camp fighting the death eaters when they appeared, they just saw it, same as the rest.
The MoM probably cannot trace the use of Unforigavble Curses because there is no wizarding trace left on you once you become of age. The only way they can detect spells is by using 'Priori Incantatem' which would show echoes of spells recently used, however, you would have to be able to luckily find somebody that had used the spell/s, take their wand from them (I doubt you would see Bellatrix lending her want to the ministry to have it checked) and then they would know. Apart from that, they cannot.