It's stability and government would depend on its neighbors. With a Bolshevik Russia sitting above it, it would have to go authoritarian to effectively combat the Reds. I think it could survive as an independent entity, in much the same was inter-war Estonia, Latvia and Lithuania survived, but it would be much-much harder. Regardless of which side wins the Civil War in Russia, the (false) notion of Ukraine being Russian would have be much more pronounced than the designs either the Reds of the Whites would have on Estonia (for example). The bonds of history and the region's resources would have been too much for any imperialist Red or White to overlook. But I think an independent Ukraine could have survived in such a situation precisely because of the resources it had available to it. I am not sure how much it would be able to use those resources and how much would have be sold off or rented out to various foreign powers to maintain its independence, but it could be done.
Emerging from the trauma of a Civil War with a permanent threat situated above it and with ethnic rivalries along its borderlands, Ukraine would not have been a happy state. Galicia would have been an open sore with a toxic mix of nationalism, religion and historic grievances thrown into the lot, while the East would have seen an anxious Russian populace looking for signs it is being neglected. There would have been a tremendous pressure on the part of the government to be seen as Ukrainian and not just Not Quite Russian. Any attempt at legitimate democratic elections and a legitimate democratic government would have been undermined by the threat of a civil war and invasion. Any serious stab at democracy would result in massacres and voter suppression on a scale more akin to a failed Central African state. If it is understood your tribe wins if it has more people, then without a strong democratic tradition and rule of law, it means you should make sure less members of the opposing tribe get to vote. There was no democratic tradition in Russian Imperial history and while some would see Cossacks as representing an anarchic-democratic tradition, by the 1900s that anarchy was wholly divorced from anything pertaining to governing. It was understood that a Cossack was free in the same sense a Wild West cowboy was free, but it did not mean the Cossack was free to vote in a government or select his leader any more than a cowboy had a say in who would be the land baron in his state. The leadership of the Cossacks was as rigidly hierarchical as the British Royalty and had more complex traditions to boot. Yes, there was a romantic Wild Field notion of true freedom and yes, Makhno was out and about, but Makhno represented a sense of communal anarchy not a democratic republic. I see Ukraine edging towards a populist conservative government, one that pays attention to the moods of the populace, but is not held in check by anything resembling a democracy as I would define it.
Ukrainian government would on paper look centralized, but given the local passions, it would not be an authoritarian state in anything but name, even if it tried its hardest. Unless it was absolutely tone deaf, no Directorate in Kiev would try to make Odessa follow its rules to the letter, never mind Donbass. What happens next depends on which side won the Russian Civil War - Whites or Reds.