Let's look at the prototype used in the movie, sierra #3. She's a stately and handsome 4-6-0 built in 1891. Her somewhat stubby 56" main wheels, known as "drivers" limited her actual top speed to 70mph. At the top end of her operational speed the valve gear barely has time to cycle steam through both sides of the steam cylinders. Often in steam locomotive design speed was limited by wheel diameter and cylinder design. I'm sure some high drivered 4-4-0 "American" type locomotives were capable of 88 mph in 1885. In 1893, the modified New York Central 999 pulled an express train at 102 mph. The other main limiting factor in speed in those days was track maintenance. Rich roads like the NYC, or Pennsylvania had meticulously maintained track specifically to maintain high speeds. The backwater Sierra with its hilly grade profile, historically famous sharp curves (read up on the Angels branch) and questionable track maintenance and track geometry would limit most revenue service to less than 40 if not 25 over most of the route. The steel rails themselves of the 1885 era were very spindly, often 40lb per yard, while big mainlines would have been laid with 80lb per yard steel rail. To the average reader this might not mean much, but what it means is the sturdiness of the rail dictated how much weight and lateral force you can put on the rails without them breaking gauge.

To go down your list of questions, yes. Britain in particlar was speed obsessed. Deeper inquiry into this will reveal a speed war between the main RRs of the time.

In 1895, the LNWR used their 2-4-0 "Hardwicke" to break the 90mph barrier on August 22nd 1895.

I have more information to share, but I'm on mobile.

EDIT: Thanks for the gold! Have a half an hour of GORGEOUS video of the Sierra Ry's #3 in action as thanks! https://www.youtube.com/watch?v=IeYYs92yN3A

Is there a way I can get a "North America- Railroad History" flair on Ask Historians? I'd love to field more questions about RR subjects in the future.

Two major safety innovations just prior to this time were the Westinghouse Brake and the Interlocking.

The Westinghouse Brake (patented 1868) was an early example of a "fail safe" system. Any separation/failure of the positive pressure air brake system, including decouplings or derailments, resulted in the application of the brakes, helping prevent runaway trains.

The Interlocking (early patents 1850s and 60s) helped to prevent collisions by interconnecting switches and signals and applying rules to them so multiple trains couldn't occupy the same stretch of track at the same time. Interlockings can be thought of as very early analog computers.

Follow-up question: how did people even measure the speed of something like a train in the 19th century? Were the engines equipped with speedometers? Were people outside of the train able to accurately measure the time the locomotive took to pass a couple of poles or something?

The faster the speeds trains achieved, the more imperative it became to find safe ways of stopping them.

At the close of 1885, there were a few different types of brakes in use:

1. Continuous, or air brakes. Braking is accomplished by the use of compressed air through a train pipe.
2. Independent, or buffer brakes. Braking is accomplished via manual mechanical brakes.

Vacuum brakes were also invented around this period of time (1872), but they were never very popular in the US. Vacuum brakes worked on the principle of creating vacuum in a train pipe to apply the brakes (these could fail catastrophically if there was a leak).

I won't talk about the manual braking systems, because the downsides to that system should be fairly obvious (inability to apply brakes simultaneously).

In 1869, the first Westinghouse air brake came into service. It was a "straight air brake," which meant that compressed air from a storage reservoir in the locomotive was used used via a train pipe to apply the brakes via a brake cylinder in each car.

There were problems with this system:

1. If the brake hose burst, or the train broke apart, all air pressure would be lost and you would lose the brakes.
2. It was slow to apply or release the brakes.
3. The brakes were applied to cars nearest the locomotive sooner than cars farther away, causing pileups. This made trains longer than 10 cars impracticable.

It wasn't until 1872 that Westinghouse invented the automatic air brake. The major innovation this introduced were separate auxiliary air reservoirs in each car, charged by the train pipe and controlled by a triple valve.

If the air pressure in the train pipe is higher than that of the auxiliary reservoir, the triple valve releases the brake and charges the auxiliary reservoir. If the opposite is true, the brake is applied via air pressure from the auxiliary reservoir.

Thus, if the engineer applies a service brake pipe reduction, a hose pipe bursts, or the train splits, air pressure will be reduced in the train pipe and all brakes will be applied via air from the auxiliary reservoirs.

There is, of course, a lot more, but I'm not an engineer and my ability to understand some of the descriptions and mathematical calculations in the source material is limited.

Derr, Louis. "Air Brakes." Cyclopedia of Engineering, Volume III. Chicago: American Technical Society, 1918. Print.