Whoa, tough question to answer. It seems to vary a little bit between the Ptolemies, Seleukids, and Antigonids - especially in terms of who is responsible for providing his own equipment or what is given to him by the state. I can't remember where I read it in The Seleucid Army, but I believe that with the Antigonids you could be fined for not carrying certain forms of equipment. So then, what about professional vs feudal - well, definitely professional with help from levies.

You could consider units like the Pezhetairoi, Thureophoroi, Thorakitai, Hypaspistai, and Hetairoi as professional. Other units liked the klerouchikoi phalangitai and prodromoi would be given land grants and would be brought on to fight when needed. Although, in the Seleukid military prodromoi seem to have been called into service for quite a long time so it is definitely possible that a fair number of them became full-time soldiers.

As far as the elite pikes go, with the Ptolemies they were given huge land grants and loads of cash. The Seleukids seemed to have instead used the most promising sons of military settlers to serve in the argyraspidai for a certain amount of time; same for the Hetairoi.

So... that is sort of the short of it. I hope that isn't confusing because I'm sure I made a mistake somewhere which Paullus or Urnamma will correct later.

And what makes you think that they didn't have the advanced administrative support that the Romans had? If anything, they had a superior system due to all the cavalry, elephants, and dependents that would travel with the army and often do so for many, many years at a time.