To answer the question in your title, not necessarily. Becoming a slave was not always permanent condition. Many Greek slaves were manumitted after a period of service, but there doesn't seem to have been a customary time to do so at any point in a slave's service. Slavery was also vastly different depending on the type of slave.

If you were a slave and had training for skilled labor, such as being a smith before the war, then you would have a relatively high position for a slave. It was often the case that skilled slaves such as these worked for (possibly alongside) their masters. Sometimes they lived in a separate home than their master and paid a portion of their income to the master, but were otherwise free to conduct their own affairs. Slave could work towards buying their freedom in this method and slave owners were encouraged to set up this kind of arrangement with a skilled slave. If the slave knew when they would be freed and it was a real possibly then he was less likely to run off and the slave represented a significant investment.

In the later case of working with the master, this allowed for a personal relationship to develop that could lead to freedom. However, when one stopped being a slave, that did not lead to an end of obligation for the master. Slaves could be freed with the condition that they still perform tasks for their old master and if these obligations were not performed that they would be re-enslaved. So if you were a skilled laborer who could work for his freedom or build a relationship with the master then your chances of freedom increased, though perhaps the freedom would not be enough to travel home.

It was considered inappropriate to mark slaves, especially Greek slaves, so the possibility of a runaway is significant in your scenario, as our Greek slave has a job to come home to and knows Greek language and custom. Any Greek slave could hope to see their family in that sense. Foreign slaves would not have that luxury, they would have nowhere to return to (if their lands were conquered) and would have difficulty blending into Greek society and surviving on their own without knowing the customs or language.

On your last point. There are great differences between the African slave trade and between classical slavery. The easiest way to explain how is to point out that the African slave trade to the Americas was racially based and classical slavery was not. Under the African slave trade, slaves were property and nothing more. Classical slavery recognized that slaves were people (Aristotle called then animated tools or two footed beasts of burden) though the master owned them. There were laws in Athens that forbid acts of hubris against slaves. Slavers could own property themselves (and though technically the masters own the slaves who owned the property in practice the slaves own it in their own right). Slaves would wield great power through their profession (court slaves in Rome had fantastic power).

A followup question, did any of the prisoners of war from the Sicilian Expedition who were sold off as slaves ever make it back? And were the oarsmen even sold off to the mines?