Pick one spell which you have the ability to cast. Whenever you cast that spell you may apply any one metamagic feat you have to that spell without affecting its level or casting time, as long as the total modified level of the spell does not use a spell slot above 9th level.
When you cast an echoing spell, it does not disappear entirely from memory, and you can cast it one additional time during that day. No effect that allows you to reprepare or recast a spell can affect the echoed spell.
Based on these, is the following sequence of actions legal?
Pick a 5th level spell as your perfected spell
Prep and cast the perfected spell, applying the Echoing Spell metamagic feat to it. The spell is still available to cast.
Here is what I'm not sure about.
Cast the echoed spell, applying a second, different metamagic feat to the echoed spell. Specifically, I want to know if I could apply the Quicken Spell metamagic feat so that I could get two castings of my perfected spell in one round with only one spell slot.
If i understand your plan correctly, then yes.
Standard Action : level 5 perfected spell + echoing spell
Swift Action : level 5 perfected spell + quicken
The second casting would, of course, consume the slot.
Also, you may want to keep in mind (and somewhat unrelated) the casting time of the spell in question.