What Roman legal structures from the Republic lasted into the Empire and which ones were evaporated? I'm aware that the senate remained (although I'd appreciate details on exactly which of its powers were curtailed).
Technically most of the political, religious, and governmental institutions lasted well into the age of empire. They existed, it's just that all of their powers had been usurped. These organizations were watched carefully and activities directed by Caesar Augustus. Augustus who was able to establish a legal framework for his absolute power, promoted and coexisted with the offices and institutions of the republic. He chose to promote these institutions as a facade to placate and distract those who would oppose him.
When the second triumvirate came to power. 47 BC, (Augustus, Mark Antony, Lepidus) they slaughtered 300 Roman Senators and 2,000 members of the class below the senators, the equites or knights. The Roman Senate and trappings of the republic were thus ceased.
After about two decades Augustus would come to power in his own right as we know today as the first Roman Emperor. Augustus had witnessed what happened to his great — uncle and adopted father Julius Caesar when Caesar consolidated all political authority under himself (dictator perpetuus), he was assassinated.
When Augustus came to absolute power 27 BC, he did so clandestinely. Rome had been ruled in practice by strong men for decades (first and second triumvirates). Rome had also been ravaged by civil wars for many years. Augustus brought back the institutions and political structure of the Republic and publicly turned the government of Rome back to the Senate and people.
Only under Augustus these institutions were a facade. The absolute power would be vested in himself. Augustus over years created a legal framework in which all real power was vested in himself. He controlled the military, was immune from prosecution, held multiple high offices for life (including supreme military commander, tribune and censor) and was able to dictate policy regardless of what opinion the old republic institutions members held. Rome's citizens and leading families were given the facade of their republic, but now they lived under one man's rule.
Augustus Government and Administration
From 31 until 23 BCE the constitutional basis of his power remained a continuous succession of consulships, but in January 27 BCE he ostensibly “transferred the State to the free disposal of the Senate and people,” earning the misleading, though outwardly plausible, tribute that he had restored the republic. At the same time, he was granted a 10-year tenure of an area of government (provincia) comprising Spain, Gaul, and Syria, the three regions containing the bulk of the army. The remaining provinces were to be governed by proconsuls appointed by the Senate in the old republican fashion. Octavian, however, believed that his supreme prestige—crystallized in the meaningful term auctoritas—safeguarded him against any defiance by these personages; and he was indeed able, more or less indirectly, to influence their appointments, just as he was able (on the rare occasions when he regarded it as desirable) to influence the appointments to the consulships and other metropolitan offices that continued to exist in “republican” fashion.
With the aid of writers such as Virgil, Livy, and Horace, all of whom in their different ways shared the same ideas, (Augustus) showed his patriotic veneration of the old Italian faith by reviving many of its ceremonials and repairing numerous temples.
Specifically what Augustus 'brought back':
- Roman Senate,
- the executive magistrates,
- the legislative assemblies.
What Augustus did not bring back:
Augustus rejected all monarchical titles and instead called himself, Princeps Civitatis ("First Citizen"). He did not live in Rome nor attend the senate meetings. He stayed at his estates, and directed his empire remotely and not publicly.
Augustus's strategy worked well, his rule lasted until 14 AD (41 years) the longest rule of any Roman Emperor. Under Augustus Rome's borders increased, wealthy and fertile Egypt was incorporated into the empire, the country knew peace (relatively and internally), and its economy prospered.