I recommend the documentary The Men Who Built America for an educational and highly digestible format to learn about John D. Rockefeller's life and how he built Standard Oil from the ground up.

Technically, gasoline did exist in 1885 but the know how to refine it into usable petroleum did not yet exist, and it was dangerously volatile and regarded as a waste product, often dumped into rivers or the ground and being highly pollutive. It was only nearing the end of the 19th century, after Thomas Edison's light bulb threatened the future of kerosene as the main source of lighting in American homes, and with the appearance of the early combustion engine, that Rockefeller invested R&D into gasoline, thus discovering petroleum as we know it today as a reliable fuel source.

Given that the combustion engine only took off after Rockefeller's contributions and that hand cranked engines survived well into the 20th century, I would say that it would be unlikely to find a fuel source as you are describing in 1885.

Chemist here. Depending on the level of detail you want the answers are yes, no(doc brown is American) and yes-ish.

So lets go through them. The the short answer is yes. Various petrolium distilates existed and some of them could run a gasoline engine. In fact the first car with an internal combustion engine was built in 1886.

No (Doc Brown is American). If he goes looking for gasoline anything he gets is unlikely to run his engine. But he's crazy enough to use a British car. Perhaps he was crazy enough to go to Britain and pick up the name petrol. Or for the time period in question Carless, Capel & Leonard's Petrol. Get hold of a few tins of that and you have something that could run a car.

Yes-ish. The problem with Carless, Capel & Leonard's Petrol is it going to have a pathetically low octane rating by modern standards. Worse still there wouldn't be much in the way of good antiknock agents available. Blending with ethanol is an option but I'm not sure that would be a good idea with a low octane fuel. Tetraethyllead while the chemistry existed to make it wouldn't be commercially available for some decades. How the DeLorean's PRV engine would handle such low quality fuel I don't know but I can't imagine the result would be any good.

For more technically specific scientific explanations, you might try /r/askscience. I can say, however, that in the United States by 1885 gas lighting was quite popular and widespread, though it was beginning to decline in favor of electricity. Though not all frontier towns likely would have had gas light availability, most prosperous ones would have.

Gas lighting has a relatively long history prior to the nineteenth century, as coal miners noticed that flammable gas would be released during the course of their operations. Later, a liquid gas was refined as a byproduct of the coking process. Once people realized the possibilities of bright nighttime light in both the indoors and outdoors, conversion of solid goal to a flammable liquid began on an industrial scale, largely replacing whale oil as fuel for illumination. Most towns of any size began to have a "gasworks," which was a central location that distributed this gas throughout the city, powering lights. Literally, it would come in jets out of the walls. You might ask if this caused fires, and the answer is "yes," but lamps, candles, and other more primitive lighting sources already caused fires, and gas light was seen to burn brighter, "purer," and more cheaply than these earlier methods, so the risk of fire was worth it. I know that later in the century a competitor to coal gas emerged in the form of a gas manufactured from crude oil, though it was never as dominant in the market. Coal gas was widespread for home use in the United States well into the twentieth century.

I would be interested to hear from someone who knows how feasible it would be for a 1985 DeLorean to be converted, using late nineteenth century technology, to the use of coal gas instead of gasoline.

For a thorough and entertaining history of the development of artificial light, I recommend Wolfgang Schivelbusch's Disenchanted Night.