A disease with an incubation time of 18 years spreads, unnoticed. 18 years later, its timer goes off and kills the majority of the earth's population, leaving only those under 18 and not infected by the strain. This leaves children to learn to survive entirely on their own. What might this society be like?
How would they cope with extreme loss without adult support?
They'd go through all of the stages of grief and undergo classic crowd psychology responses to disasters (10% panic, 80% freeze, 10% act). This being said, kids have a remarkable ability to adapt because they are young. For inspiration I'd recommend looking into the psychological and sociological development of children and minors who grow up without parental figures. Similarly, emancipated minors.
Where would they reside?
This will vary dramatically. I expect close knit friends would crash at each other's place (forming familial groups). Any 17-18yo in Student Government would likely try to setup a common shelter for everyone at the school. Delinquents would sleep wherever they want, likely places they weren't allowed to before (as a minor) as an act of rebellion.
How would they come together to form a new society?
Given their age, I expect a simplistic government would form out of school hierarchies. Minors who participated in Student Government or Model UN have the diplomatic and bureaucratic know-how to form and maintain simple government. Anyone from honors and preparatory schools would have a leg up in become subject matter (if not de-facto) experts in respective fields. With the loss of adults, talent would be recognized and exploited (or supported) rapidly. Similarly, their will never be a shortage of jobs, so 'unemployment' may be considered eradicated. Additionally, consider cliques. Within a school hierarchy, factions will form rapidly from cliques.
How would they get food and clean water?
Literally every hypermarket, and grocery store: Wallmart, Costco, Wholefoods, TraderJoes, etc. Locally for your area: Brookshire Grocery Company, Cannata's Family Market, Lakeview Grocery, Mac's Fresh Market, Market Basket, etc...
Anyone near farms have direct access and likely knowledge to grow their own food. Similarly, they can easily form a trade community or become exploited by larger communities.
Anyone with technical know-how, may suggest filtration devices, which may be obtained from any of the hypermarkets listed above.
Would they figure out how to harness electricity? How?
Yes. Anyone who has a generator already knows the basics: put fuel in, press a button, electricity. At a grander scale, anyone with an interest in electronics or electrical systems has libraries of knowledge at their disposal. I suspect if it becomes an issue, schools may devote joint resources to maintain existing electrical structures; however, they'll run into issues with respect to the size of utility companies. Additionally, there will be edge cases with access to renewable energy (hydro, wind, solar), in which case, their requirements would be much relaxed, but their electrical draw highly limited.
In particular I'd be concerned for the order of failure of large scale systems. Local and Oil will automatically shut down when they run out of fuel. Nuclear systems, if automated, will automatically insert rods when the reactions get to hot; but, depending on the software, may either shut down the reaction for good, or attempt to maintain it indefinitely. Either way the nuclear plants will slowly lose potency. Given your region (which I'm not familiar with), I wonder if there are any hydroelectric sources. They'd be capable of producing electricity for a few years without maintenance. After that though, they'd require major maintenance upkeep to maintain capacity.
How would their society advance if there were no adults to teach them?
That's simple: a disease wiped out all adults but left the infrastructure entirely intact! Every single library, repository, and computer is up for grabs. All of those script-kiddies and amateur programmers now have free reign across the internet. Every bookworm has access to libraries without late-fees! The minute an interim government forms, a demand for knowledge will predicate the need to obtain and maintain these knowledge centers.
What might their self-government be like?
As I stated before, I suspect the SGA geeks would attempt to form a government; however, they will be competing against oligarchical structures forming around delinquents and individuals who seek dictoral control. On a larger scale, we'd see all three forming. The question then becomes: which ones succeed faster? My expectation is that the delinquent run oligarchies would rapidly form classic raid parties in post-apocalyptic scenarios. The tyrannical and SGA democratic communities would recognize the need for forming communities and putting talent to work on rebuilding and maintaining said community; but, the exact labor structure will vary greatly. I can see any number of labor structures rising: serfdom from cliques, technocracy in city communities, etc. Secondly, they would also come into conflict due to differences in ideology and diplomacy. How they handle it is up to you.
How would those "in charge" assert control?
How do children typically assert control? Bullying and Humiliation, or through sheer numbers and popular support. If a delinquent or tyrant gets their hands on a gun, sheer terror will reign until they are disposed. In more democratic settings, sheer popular support may keep tyrants from gaining power.
More importantly I suspect, is guns. The minute someone gets smart and amasses weapons, they immediately have the power of life over death of their peers. Depending on how they choose to distribute this (if at all) any variation of tyrannical or democratic power structures may form.
How would they deal with simple things such as toilets and laundry?
That depends: how long will those infrastructure remain active without supervision? And how long will toiletry supplies last?
Would they revert to a previous time period?
I doubt it considering all technology and infrastructure would remain untouched. Give the 18yo 5-10 years of self-studying and on-the-job learning w/o supervision and I'm confident they'll be able to reclaim much of our technology and infrastructure.
I think this also provides an opportunity for you to write about how children will act without the threat of adult consequences. Sexist and patriarchal men will rape women. Bullies will unleash holy hell without due regard for consequences. Victims would outright murder those who injured them Children would run amok until someone gets hurt. 17-18 year olds will find themselves thrust with the responsibility of adulthood when they were only recently looking forward to college (studying, travelling, or partying).