I think this is widely under-rated. Lincoln was the nominal head of a party which was filled with rivals, and which hadn’t entirely settled on its platform. The conflict between abolitionists and no-nothings alone could have paralyzed the party. In addition he was constantly dealing with political issues coming from the various wings of the Democrats.
Lincoln masterfully kept the Democrats divided and the Republicans mostly united. He maneuvered his rivals either out of the picture (basically he provided all the rope needed for Cameron to commit political suicide), or over to his views (like with Chase who did it for political standing, or with Seward out of eventual genuine respect and friendship). Lincoln had virtually no decision to make which wasn’t a political landmine. And yet somehow he kept all the mules yoked together and pulling in the same direction. It was a masterful accomplishment, without which the war couldn’t have been won.
The Apache wars started before he was president and continued after. So there was a state of constant but low level war throughout the southern plains and the Southwest for the entire civil war. And the biggest battle of the western Indian wars, First Adobe Walls, took place in 1864. But this wasn’t really a result of Lincoln’s policies. Mostly it was fought by impromptu bands of settlers and miners and local militia rather than federal troops. And outside a few large engagements, they were more a series of raids from both sides.
The Navajo wars mostly took place during the Civil War. And again mostly they consisted of raids rather than war. And mostly it was white civilians doing the raiding not actual troops. Really the whole thing reads more like the Hatfields and McCoys rather than a war. For example after a treaty was signed, in August 1861 there was a horse race attended by both the Navajo and local militia. Accusations of cheating broke out and it turned into a firefight… and the wars were back on. By 1863 a reservation was established. From 1864 through 1866 most of the Navajo were tracked down and forced to take “the long walk” to Bosque Redondo.
The Mormon-Ute wars had died down a bit for most of the Civil War, but kicked back up just as the Civil War was ending. It is really hard to connect this one to Lincoln at all though. It was a result of drought in the area and regular hostilities between the Mormon settlers and the Utes, rather than any federal policy.
The Colorado War took place entirely during the Civil War and like the others was mostly started and prosecuted by white settlers. And nasty as it was, it can’t really be laid at Lincoln’s feet either, as again it was fought by local volunteers.
In comparison to these the Dakota war is the one that really was Lincoln’s Indian war. It is the one where the cause and the prosecution of the war can be laid at Lincoln’s feet. Of course the Dakota’s complaints go back further than Lincoln’s administration. But unlike the other wars, it really was the Feds that provided the casus belli. As part of their treaty obligations, the US was to provide food and money to the Minnesota Dakota. By 1862 they were over $1 million behind in payments. So when one band went to get food from the Indian agent and were refused they went to war. Hundreds of settlers were killed and eventually General John Pope was sent to fight the war. He ended up with about 10,000 troops and ended the war pretty decisively. All the Dakota reservations in Minnesota were abolished and all the Dakota were forced to relocate to the Dakota Territory.
In an unusual move after the war the local white settlers rounded up Dakota men and put them on trial for murder and rape. About 300 were convicted in “trials” that were only a few minutes long each. Lincoln intervened and commuted all but 39 of the men. While the war was officially over, thousands of Federal soldiers remained in Minnesota. And in 1863 and again in 1864 the army fought campaigns against the Lakota and many of the Dakota refugees in the Dakota territory.