I've been under the impression that legions were equipped and paid for out of pocket by the consul/proconsul who mustered them. Did the same principle extend to government officials carrying out civil duties?
For example, when a governor wanted to build an aqueduct or road in his province, did he need to ask the Assemblies back in Rome to release tax money to him? Or did he use personal funds/taxes gathered from his own province? Same question applies to building temples, stadiums, and other public works in Rome itself.
When a general enriched Rome by conquering new lands, who was specifically being enriched? Legionaries carting home looted goods/gold/slaves? The general himself carting home LOTS of looted gold/goods/slaves? Or did the general cart the loot home and hand it over to some government body for storage?
When a client king paid tribute to Rome, where did his money go? To a big pile of gold in a vault under the Senate building? To the general who conquered him in the first place? Who gets the money if the general has died of old age or been exiled/disgraced by a political scandal? Same question applies to fully integrated provinces.
Who pays for the bread and circuses? Where is he getting that money and who is he giving it to?
The best info for this is found in Polybius 6, an extended discussion of the Roman constitution and army.
The Senate controlled Romes finances. Polybius specifically uses this as an example of how the Senate has a check over the consuls and other magistrates. A consul may want to make war, but he relies on the Senate to supply his army.
However, in actuality there was a lot a magistrate could do once he left Rome to maintain his independent initiative. Some recruited armies out of native peoples (it happened a few times in Spain, for example), and they could try to support their armies through local taxation, pillage, or scavenging, although the latter two relied on being successful in war.
Public games were paid for in part through public funds but increasingly they were used as a way for magistrates to demonstrate their generosity. This is a big aspect of Cicero's argument in Pro Murena.
The money was kept in temples. Caesar began his civil war by marching to Rome and stealing the money from the Temple of Jupiter, for example. The idea was that the gods would protect the money and discourage theft through threat of divine punishment. Also religion was a civic institution in Rome that was entwined with government so it wouldn't seem all that odd.
Taxes and tributes were gathered by contractors called publicani. They would bid for the contract by claiming that they could get so much money from a particular province. The highest bidder won the contract then collected the money under the authority of the Roman pro-magistrate. They supported their enterprise by keeping any money they taxed beyond their bid (hence they had a reputation for ripping people off in the New Testament).
Most public works were undertaken by cities, not the empire. Each city/region mostly governed itself but had access to certain services the empire could provide (defense of the army, appeal to magistrates in legal affairs). The local council would handle infrastructure mostly through personal funds, again to display generosity for a stab at popular favor. Temples, theaters, and other public works often bear the names of the men who built them. These were done, like games, with a mix of public and private funds.
The exception to this are the roads, many of which were built by specific legions for strategic purposes. However, most roads were built on a local basis, including the major roads to and from Rome which had been around for ages, for example the Appian Way, named for the man who built it as censor.