So what's actually going on in Ukraine with the fighting? Like, how are the battles being fought? How has it lasted this long? Why isn't the government's military able to defeat the separatists? They're obviously backed by Russia, but since Russia is still trying to maintain some plausible deniability, the backing shouldn't be enough to let the separatists manhandle a government army a million man strong... so what gives? I'm under the impression only a fraction of the Ukrainian military is mobilized and fighting. Why? It seems like all the fighting is super half-hearted. Got to be, or it'd be over, one way or the other.
Ukrainian army is currently blocked in its actions. Ukrainians can't use artillery as actively as it's needed because pro-russian forces are shooting from cities, that would mean a total annihilation of everything inside cities including civilians. If a single explosion in the city which takes several lives of civilians gets such a huge social coverage in media imagine what would happen if you widely use artillery with that capacity of troops on such a small territory (nothing to say about thousands of dead civilians each day if not hour of such usage of artillery). Same things go for air forces and so on. Basically pro-russians are having unlimited amount of everything, it's like playing against a cheater, you kill 10 of his tanks and 300 of infantry and on the very next day he gets new. Backing by Russia can have any amount of troops/weapons. At this point Ukrainian military system works a lot on its reserve personnel to be ready for full scale invasion, also time works for Ukraine now as Russia suffers economical crisis by now. If it was up to me - I would leave the whole situation there as it is, live of people are not worth it. Also people need time to understand and compare.
As Vovka stated correctly, Donetsk Airport was not a strategic place, rather a political. There was no strategic sense to hold it as neither russian nor ukrainian aircrafts could land there, you need just 1 day of artillery work to make it impossible to land. Ukrainians held it until everything got destroyed but pro-russians on their hand didn't actively try to capture it during all those 200 days, was more like attempts with breaks.