Some points to maybe think about finding ways to incorporate:
Sparta's culture was very conservative, almost fascist (just being honest). Society was highly stratified, with the upper echelon jealously guarding its power over the lower classes. It was so difficult to enter into the upper class of society, that the number of Spartans in this class dangerously dwindled over time. By the time reforms were implemented it was too late to save the traditional order.
The entire state was designed to maintain their army of Spartiates so that they could keep their population of serfs under control. Once a Spartiate became a man, he was given an estate which came with a population of subjugated serfs (helots) who worked the estate and supported the Spartiate's life of military training (it was dishonorable for a Spartiate to work the land or work to earn money).
Women exercised like the men did, because strong mothers can bear strong sons. Strong sons can grow to be strong soldiers for the state.
Spartans did not use gold or silver currency, except to pay foreign debts (like hiring mercenaries). Internally they used an iron ingot as currency or simply bartered. Trading and other merchant activities were carried out by the Perioikoi class, the "middle" class of Spartan society, who were obligated to defend the state as hoplites, but otherwise held no political power.
The Agoge to train Spartiates from a young age was real. It had age classes and lasted until early adulthood, at which time they were required to marry a Spartan woman and begin to sire children. Spartan men who failed to complete the Agoge or who otherwise had become disgraced fell into a type of "limbo" class if landless pariahs. It was possible to regain one's honor and status, but rare.
Spartan foreign policy largely revolved around building a series of alliances as a buffer against potential enemies, culminating in the Peloponnesian League. Sparta was the clear leader of this league and compelled its members to supply troops to Spartan armies when necessary. The benefit of joining the league was stability and the protection of the Spartan army. Functionally, this was a protection racket. Spartans did not like to send their army too far afield, lest their subjugated populations and allies take the opportunity to revolt, which did happen on a number of notable occasions.
Sparta was ruled by 2 kings, each of a specific family line. When one king was leading the army to war, the other king stayed home, so that there would always be at least one king in Sparta. The power of the kings was checked by a small council of elder Spartiates called ephors. They had the power to impeach a king for impropriety, cowardice, corruption, or blasphemy, and they exercised this power frequently over the course of their recorded history. And under the ephors was an assembly of Spartiates having reached middle age.
The city of Sparta, like early Rome, could more accurately be described as a collection of villages scattered around a common acropolis. The city sat nestled in the middle of a long fertile plain flanked by two mountain ranges. It famously had no stone walls until Hellenistic times, but did have a palisade on occasion of national emergency, once famously built by the women of Sparta after their men were defeated by Pyrrhus of Epirus.
More to come.