Regulation

Yes, once elastically invariable alloys for hairsprings had been developed, collectively known as "Elinvars", the temperature compensation, which had been principally to allow for the changing elasticity of the hairspring, became much less important, and un-split ovalising balances, much less fragile, became feasible. There are also balance wheel rims made from laminated Zinc, which expands differently along and across the "grain". Possibly the first compensation system developed (on the wheel itself) was by Breguet, who placed two mercury thermometers on the wheel rim with their expansion tubes running towards the centre (!).
"Anti-magnetic" Elinvars also prevent the spires of the spring from sticking together through becoming magnetised, which could cause the watch to sprint through its day in 14h!

Rate in watches is most commonly adjusted by varying the effective length of the hairspring, this passing between two "curb pins" mounted on a lever concentric with the axis of the wheel. This has both advantages and disadvantages - neither balance nor pendulum are strictly isochronous, smaller amplitudes taking (slightly) less time. By allowing a small amount of play between the curb pins, at lower amplitudes the hairspring behaves, during the time when it passes from one pin to the other, as if it were longer, thus increasing the period to compensate.

In some Chronometers there are smaller screws which, when removed, increase the rate so that the watch runs on Sidereal Time, which is measured by the rotation of the Earth with respect to the fixed stars. Because the Earth moves in its orbit about the Sun, the sidereal day is about 4 min shorter than the solar day. (23h 56m 04.0905s, for the very punctual). This avoids a calculation in astro-navigation, but may mean you turn up early for breakfast. I sailed with Russians who insisted on remaining on Moscow time, French crew who would use their local (+2h in summer) and of course GMT GPS ship's time. What fun, organising watches!