Rome did not really ask for levies, at least not in Greece. Like most empires, Rome had a very strong appreciation of the diversity within its borders, and included in this was something quite like the concept of the "martial race". "Martial race" was a designation the British used, particularly in India, for peoples they considered warlike and ripe for recruitment into the military such as the Sikhs and the Gurkhas, but it is a tendency that can be seen in a huge variety of different empires--for example Agkhor recruited heavily from the "warlike" Thai hill people, and the Ming recruited heavily from the Miao in the southeast. The Romans, likewise, relied disproportionately on individual groups, such as the Batavians and the various Berber people, to fill their army, and did not recruit as much from people they thought to be "soft". And while the Roman army was technically volunteer, the sort of tactics and coercion used to recruit from the "martial races" certainly qualify as "enlistment" at least.

Taxation was assessed by surveyors and carried out by the community. So basically the Roman surveyors would say "you owe x" to the local government, and they would be in charge of gathering it. Some communities were exempt from taxation for honorific reasons.

For permanent presence, I suspect the most common way the Roman administration would be felt would be the court system. Local communities would have their own methods of settling disputes, but if they proved intractable, they could be handled by the governor in a system similar to the Medieval assize courts. Some communities however, would have quite a lot of interaction with Roman authorities and feel very connected to the imperial center, others were probably more or less neglected. The extent to which someone in Athens or Corinth interacted with the Romans and the extent someone in, say, the inland Peloponnese did would be very different.