How do my fellow Spartans perceive me?

All male Spartan citizens had to go through the agoge, with the sole exception of future kings from one of the two Spartan royal houses. Your fellow Spartans would probably find nothing special about you.

During your education, you were the responsibility of the paidonomoi ('boy-herders'), but now that you've completed the agoge you have to be voted in to one of the tent groups in which Spartans spent their days in peacetime. This means that your fellow Spartans will now see you as their responsibility, and you're kind of a burden, because you're just a young whippersnapper who has much to learn. The agoge has not taught you how to fight; it is the tent group that must turn you into a warrior. Only a unanimous member vote allows access to one of the tent groups, so you had better hope you've made a good impression in your education so far. If you're not voted in, you'll have to spend a year in limbo before you can try again.

If you've not yet made a name for yourself, this would be a great time to do something conspicuously Spartan that would make older citizens proud to take you under their wing. Displays of selfless obedience, stoic indifference to pain or hunger, or extreme endurance might do the trick. Better yet, win an athletic contest. Best of all would be to challenge a member of the elite hippeis to a fight and beating him. That might actually get you selected for the unit, which is the highest honour you'll probably ever get. But more likely you'll get curb-stomped to the amusement of all those present.

Was this training indicative of a higher moral capacity?

The Spartan upbringing focused entirely on instilling Spartan values in Spartan boys - obedience first of all, but also endurance, indifference to hardship, contempt for pain, and fiercely competitive ambition for honour. The Spartans believed that this made them both morally and physically superior to other Greeks (though you, as a fresh graduate, would probably not stand out among them unless you let a fox gnaw its way through you without flinching or something juicy like that).

Other Greeks would not be so sure that the agoge made you a better man. The Athenians believed that your unflinching obedience came from fear of the lash; you've been conditioned in such a way that you can never know true courage. Your training has seized your creativity and your curiosity and ripped it out root and branch. You have been taught to love Sparta and despise the rest of the world. You are now just another cog in the Spartan war machine, destined to stay aloof from all the things that make life worth living. Your education lacked a lot of the sophistication - the poetry, the philosophy, the rhetoric - that makes a leisured existence worthwhile. What do you know of morality, when you barely know how to think for yourself?

Would I be eligible to participate more in society if I had completed this training?

You are now eligible to join one of the tent groups (syssitia) that make up the society of male Spartan citizens in times of peace. You are also eligible for military service. When the ban is called out, and your year-class is included (as it usually will be until you hit 40), you will have to march out with the rest of the militia to fight in Sparta's wars. If that happens, you had better learn to get along with non-Spartiate Lakedaimonians, because they're going to be making up the lion's share of your phalanx, and they don't give a toss that you went through the agoge.

Also, you are not yet a full-grown adult. You're still young (just 20 or 21), rash in your actions, quick to judge, and prematurely proud of your achievements. In another ten years you will finally come of age; you'll be allowed to take a wife (whom you might have to share), take part in assembly meetings (if they are called, which rarely happens), and try to get promoted to officer (though the job's probably taken by those who got to take part in the krypteia or those who served in the hippeis, because they're better than you).

tl;dr: you're a Spartan, your life is shit