Different methods would have been used by different people at different times, as one might expect. Sometimes they used the carrot, sometimes they used the stick, and any successful conqueror in the classical period used the right combination of both. They also needed loyalists in place everywhere they went.

I'll talk a bit about Rome, but since you used the example of Alexander I went and found an article on this topic for you. Alexander took a personal hand in the administration of his conquered territories, but delegated responsibilities whenever possible to trusted subordinates in areas such as tax collection (168-9), backed by Macedonian military power wherever opposition appeared. In his initial forays into Persian territory, Alexander would appointment friendly satraps in place of the Great King's appointments (171). In Egypt he tactfully placed Egyptian "nomarchs" in charge of Upper and Lower Egypt, and though Macedonians controlled the army (171), Egypt was ruled according to local customs. Alexander was apparently pretty pragmatic in his arrangements: when the Greek Cleomenes took over Egypt as satrap, Alexander left him in place as he was clever, wasn't a threat and he could turn Alexander a nice profit. Once Alexander was able to see that he could take over the entire Persian Empire (made clear at Issus), he adopted the Persian policy of appointing loyal (Iranian) satraps, while take control of tax collection and the military with his own picked men (173-6). However, as he did not capture Darius at Gaugemala and he burned Persepolis, Alexander lost his legitimate claim to kingship and the loyalty of Iranians became more and more difficult to keep, so he began appointing more Macedonians (176-7). After Alexander's return from the abortive Indian campaign, he utilised harsher methods to maintain control, but maintain control he did (180). According to Badian's article, Alexander did not uproot local aristocracies, (though deaths in battle might have made an impact), kept his picked people in charge with other picked people in control of finances and the army to make rebellion difficult, and utilised garrisons, local recruitment and separate armies such as the Macedonian army used to suppress Greece to deal with major issues. I can't comment on language barriers, but I daresay a highborn Persian might have some knowledge of Greek, and Macedonians and Greeks also spoke Greek: nonetheless, Greek became the language of government.

More in my area of knowledge, Caesar's campaigns in Gaul are an outstanding example of a rapid expansion into turbulent territory, wherein Caesar had to use every method in the empire-building playbook to maintain control, came unstuck multiple times, and thanks to his memoirs we know a lot of the details. Caesar greatly valued and rewarded friendly aristocracy such as the Aedui, or appointed friendly leaders in tribes with a less friendly aristocracy, or completely obliterated the aristocracy of hostile tribes such as the Nervii or Eburones.

He did not split his army up into vulnerable garrisons unless necessary, with two exceptions. He split his army up into several smaller divisions spread all over the country in the winter of 54 BC because of logistical issues and because he trusted the military situation to remain stable enough over winter, but he was wrong (losing nearly two legions and two good generals in the process). He also had a significant detachment of troops protecting his treasury and grain supplies in friendly territory, but when the trusted Aedui betrayed him in 52 BC, he lost those troops too.

Caesar recruited many of his troops from his province, Cisalpine Gaul, though of course many came from Italy. They were basically "Romanised", and once inculcated into the legions were perfectly reliable. He also used Gallic auxiliaries from loyal tribes (until there weren't any loyal tribes, but their performance was mixed anyway), and Germans, who performed particularly well. Caesar had enough allies in Gaul that he no doubt had people who spoke the language who could help him communicate with people.

So from these two different case studies (of admittedly very different scales), you can see the answers to your questions might differ greatly depending on the context. If I were to generalise in order to answer the topic question briefly, the "rule of law" was never as important as the ability to draw income from a territory, and to ensure there was no rebellion. If the authorities, whether Alexander or Caesar, were getting the resources they needed and nothing was hindering their operations, it wouldn't concern them a great deal if the places they conquered were in a state of social anarchy, though you might expect too much internal stasis to affect operations, as it did when a faction of the Aedui betrayed Caesar in Gaul. The installation of loyal leaders and governments was always important, as was the use of garrisons or mobile military units that could defeat against any open rebellion. It is important to note that Caesar nor Alexander cared one whit if their subject peoples didn't "change much", as long as they didn't fight back and they gave what they were asked for.

Sources:

Caesar, Bellum Gallicum. I know quoting ancient sources isn't ideal but I know my way around the Gallic Wars pretty well and I'm confident I can comment on it without quoting secondary sources.

Badian, E. "The Administration of the Empire," Greece and Rome, 1965, 166-82.