Explain the roots of the Civil War in the sectional crises and conflict over slavery in the 1850s. Was war inevitable at that point, or were other resolutions possible?

The roots of this crisis ultimately can be traced back to the dramatic territorial expansion which the United States underwent in the 1840s, along with the rapid growth of California, which, even as early as 1849, was already seeking to apply for statehood. In the process, long-standing tensions concerning the future of slavery in the United States were reawakened.
Ultimately, these arguments and tensions concerning slavery evolved and escalated across the decade (and indeed, these tensions can be traced further back into the Missouri Crisis and the Missouri Compromise). Ultimately, however, you should be aware of the stakes and nature of this argument. It was at its core a Constitutional argument, and both sides were motivated by strategic calculations: abolitionists were aware that if they could gain a majority among the states, they could move to ban slavery by Constitutional amendment, and slaveowners were well aware of this fact. This question of the status of slavery within the territories posed had significant long-term ramifications when you factor in that, eventually, those territories would be turned into states.
These tensions were reawakened when California applied for statehood, outlawing slavery in its constitution, which would have upset the balance of power between free states and slave states. This initial crisis was managed via the Great Compromise, but it was a compromise neither side was truly happy with. This compromise would include the harsh Fugitive Slave Act to pacify slaveholders (much to the anger of abolitionists) and would also introduce the concept of popular sovereignty as an answer to the question of whether slavery should be legal or illegal in the territories.
This crisis escalated over the course of the 1850s. The Fugitive Slave Act would intensify tensions and outrage within the North, and the concept of popular sovereignty would create turmoil, especially when it was applied to the Kansas and Nebraska territories, with pro-slavery and anti-slavery advocates pouring into Kansas to influence the decision and engaging in sectarian violence against one another. The newly formed Republican Party would gain prominence (and the later election of Republican Abraham Lincoln to the presidency would prove to be the final straw which spurred secession), while the Dred Scott ruling would further increase tensions and divisions with its sweeping defense of slavery (and this is to say nothing about the impacts of the publication of Uncle Tom's Cabin or John Brown's attack at Harper's Ferry would have within the South). When looking at this tumultuous decade, you might get the sense that you are looking at a series of escalating moments in which these tensions and divisions continue ratcheting up towards higher levels of intensity.
Now, your major question here, with this in mind, is whether the Civil War was inevitable, and this is actually a deeply difficult question (and one which I'm honestly of two minds about). First of all, it's important to note that history is contingent. In that sense, it's hard to say anything is truly inevitable. That being said, the real question which must be asked is this: is compromise possible? To that question, I actually think the answer has to be no (or at least, I can't think of any realistic compromise that could be reached).
Ultimately, for both sides in this argument, the question of slavery was a question on which they could not afford to lose. Among abolitionists, it was a profound moral outrage. (Indeed, even if many among them may have advocated for a more gradual, more conservative path to abolition via Constitutional means, that's still far removed from total capitulation.) As for the pro-slavery side of this division, they understood slavery as part of the guaranteed rights to property and a fundamental part of Southern society. With that in mind, I'm not sure either side could afford anything less than a complete victory, where this question was concerned.