The cities where usually "free" but recognised the king as the person who granted their liberty. That's why Antiochos III wanted to take control of Smyrne and Lampsacus which had asked the help of the roman. He wanted to grant them liberty, but by doing it himself he ensured that a specific relation was made between the city and the king, enabling divine honours, statues in the bouleuterion ("council chamber") of the city or epithet (like soter: saviour). If you consider only the first generations of hellenistic kings behind the diadochoi, remember that Athens decreted an incredible number of honours to Demetrios Poliorketes, erecting an altar where he landed from his chariot and making his mouth an oracle (among other things!) Alexandria is the main exception (there was maybe others, but I don't know them) as was directly run by the lagid administration until roman conquest.

Can we compare the power of the diadochi, with that of Alexander, well the problem is that Alexander made little other than commanding armies and making peace terms with his new subject, to run a territory on the long run is a quite different thing. Remember that if the Alexander and the first generation of diadochoi could simply mint the persian gold to pay their armies, they had to organize taxation system to made their kingdom to work.

The main institution is clearly the army. Even when the army was assembled as an assembly (following the macedonian custom) it was still (the macedonian part of) the army. It played important events during the reunion of the successors around the body of Alexander in Babylon and also when Seleucos announced that he gave his wife to his son Antiochos, he was acclaimed by the assembled army as the best of the successors. The court of the kings played a huge role in the power of the kings but developped with the fixations of the different kingdoms, but the court of a king is not necessarily an instrument of power for the king himself has it is a reunion of powerfull people who could be controled, and entrusted by the king if he has enough power but can also try to control the souvereign if he lack power (due to youth for example).