We know that Yojimbo and the Magus Sisters were sent too, even though they didn’t appear in the FMV (probably for the same reason Yuffie and Vincent didn’t appear in the FMV at the end of FFVII), because they show up as Dark Aeons in the next game. We understand that each of Yuna’s Aeons can no longer be called by anyone ever again.

But that’s the thing right there:Yuna’s Aeons.

Are we to assume all of Yuna’s Aeons were the only Aeons to exist? How far does Yuna’s Sending stretch? According to the party’s travels, it seems pretty localized. As in, Yuna needs to be at least in visual range for the Sending to work at all. Otherwise Summoners could sit back at home and Send whomever or whatever they wished from the comfort of a cozy chair at home. What’s stopping Phoenix, Quetzalcoatl, Siren, Tonberry King, and others from lying in wait in some weird, dark corner of Spira no one knows about or remembers? Their original purpose may be gone, but Aeons and Summoners existed before Sin (remember, Sin was created to combat the Summoners of Zanarkand, not the other way around), and it stands to reason they could still exist after.

If the Calm always lasts around ten years, and we assume that Yunalesca created a brand-new Aeon for every final Summoning since that point, then there should be at least one hundred Aeons in existence. We know of ten percent of these: Lord Zaon, Jecht, Valefor, Ifrit, Ixion, Shiva, Anima, Bahamut, Yojimbo, and the Magus Sisters. It’s very likely that after doing their unseemly duty as Sin, the next Final Summoning simplyreplaced theminstead of destroying them. As Bevelle’s ultimate weapon (haha),  Yu Yevon using the enemy’s assets against them is an effective strategy. Sadly, it was too effective. 

It’s possible that Sin randomly (or not-so randomly) ended up destroying an unknown number of Fayths just rampaging around as Sin does, but even a fifty percent Fayth destruction rate seems kind of high. Where are all the rest? Could Yuna or someone else become a Summoner once again? Obviously, I wonder.

This is an excellent and complicated question, or rather, questions.

I. How far does Yuna's "sending" extend? How could she send the fayth when their statues were so far away?

If we push past "because PLOT," then we're left with a number of not-quite-satisfactory answers:

The aeons, through Bahamut, asked Yuna to release them from bondage. Maybe they couldn't release themselves, but a priestess of Yevon and a High Summoner had the spiritual authority to declare their fayth contracts null and void, so to speak. This assumes that Yevonite ritual magic of the sort used to bind fayth to statues is like an unbreakable vow, a spiritual binding which can only be severed by someone with Popelike authority.

Maybe the aeons were a little like unsent: they couldn't let go until their focus purpose was fulfilled. Once Sin was gone forever, they could rest, either because they chose to leave, or because, like an unsent, they were obsessed: they were psychologically incapable of leaving while they still had duty hanging over them.

Yuna had grown in power more than any Summoner before her, thanks to her pilgrimage training. The summoners pray at each temple not just to enlist aeons (which they honestly don't need, if they've got enough guardians) but also to train and fortify their soul until they're capable of handling the Final Summoning. We know that merging with ordinary fayth takes a lot out of a summoner — Yuna staggers or collapses each time she attains a new aeon, and Wakka says that summoners have died in there! So Yuna's "trained up" her spiritual powers through many ordeals. In fact, she's taken "extra credit" courses by picking up aeons outside the temple pilgrimage route: Yojimbo, a lost temple aeon, the Magus Sisters, who I suspect were Belgemine's Final Aeon (she says she was "not able to" defeat Sin in a way that suggests she tried), and Anima, who was Seymour's Final Aeon that he chose not to use on Sin. While Seymour tells Yuna back at Operation Mi'ihen "your powers are still too weak," that's early on in her pilgrimage. Later, Belgemine tells Yuna that she's stronger than her father. That makes sense: each High Summoner has to be strong enough to conjure an aeon capable of defeating her predecessor's Final Aeon. So maybe — maybe — by the time Yuna finished her pilgrimage, she did have the power to banish at a distance. 

Tidus and Jecht, as dreams-made-real by the peculiar life & death powers of Sin, could break the cycle. Bahamut says something very odd to Tidus: "We've been dreaming so long...we're tired. Would you and your father... Would you let us rest? Both you and your father have been touched by Sin. Sin, the one whom all Spira--the spiral--revolves." What does Bahamut mean? Is he asking Tidus' permission to end the dream, even though it'll mean the end of Tidus? Or is Bahamut implying that Sin is somehow a catalyst, almost like the Ring of Sauron: once it goes, everything touched by or connected with Yu Yevon will collapse. That explanation doesn't entirely make sense, because the temple fayth weren't created by Sin or Yu Yevon (Or were they? See below). But the fayth were dedicated to Sin's eradication, so I guess their existence may have been bound to it in some way.

The aeons had been in close proximity to Yuna, because she had just summoned and defeated all of them in turn. This doesn't entirely make sense either, but Bahamut had specifically asked Yuna to summon each of them so that they could be possessed by Yu Yevon, drained and defeated. She ritually killed each of them during the end-battle. Maybe that made the sending possible, even though their statues were far away. 

II. How many aeons are there? 

You said: "If the Calm always lasts around ten years, and we assume that Yunalesca created a brand-new Aeon for every final Summoning since that point, then there should be at least one hundred Aeons in existence."

That "ten years" bit has led to a lot of confusion. It's been ten years since Braska's pilgrimage ended. But the Calm, the period of Sin's absence, didn't last that whole time. Chappu died in a Sin attack a year ago, and Lulu's been on two pilgrimages before Yuna's. Braska's Calm must have ended before Chappu's death (a year ago) and Lulu's prior two pilgrimages.

Shockingly, the Ultimania guides put out by Square claim that Braska's Calm lasted about a year. That sounds crazy — who in their right mind would sacrifice themselves just to provide the world with a year without fear (on the off chance, as Braska said, that this time it would work for good)? In the real world, though, we see people sacrificing themselves for all kinds of selfless reasons: think of those monks setting themselves on fire to protest America in the Vietnam War, or Tibetan monks doing it to protest Chinese occupation. 

Therefore, in the past thousand years, there's been only a few very brief years of Calm— right after a High Summoner succeeds— and the rest of the time, it's business as usual with Sin ravaging the countryside and summoners going on pilgrimage and trying and failing to get the Final Aeon. 

How many Final Aeons are there?

 Only five times in the past thousand years did someone (Yunalesca, Gandof, Ohalland, Yocun and Braska) succeed in attaining the Final Aeon and defeating Sin. Each time, they destroyed the fayth/aeon of the previous High Summoner. So, of all the High Summoner aeons, the only one stilll kicking around in Yuna's day was Jecht.

So the question is, were there other Final Aeons which, for whatever reason, did not succeed in defeating Sin and canceling out the previous one?

We know of two possible candidates: Anima seems to have been Seymour's Final Aeon, which he refused to use in battle with Sin, and the Magus Sisters, which I think were Belgemine's. 

Again, were there any other Final Aeons we don't know about?

From what I can tell, it seems to be incredibly, incredibly rare for a summoner to survive his/her pilgrimage and reach Zanarkand. Nor is reaching Zanarkand any guarantee: a summoner has to (a) agree to kill a guardian; if they refuse, Yunalesca kills them (b) have a bond of love with a guardian that's strong enough to create a Final Aeon and (c) have the spiritual fortitude to join souls with the fayth of the Final Aeon (I think Seymour cracked).

So I could easily believe that there aren't any other Final Aeons to find. 

What about the regular garden-variety fayth, the non Final Aeons with fayth statues housed in temples?

From the Ultimania Guide  put out by Squaresoft:  "The Yevon Temples have regional chapters of temples all over Spira. Eliminating those that have been destroyed or abandoned along the way, the present temples total five in number. Though each temple has differing characteristics, the common feature they all have is a Cloister of Trials and a Chamber of the Fayth. Summoners gain aeons by undergoing the temple's trials and communicating with the fayth sealed within the fayth statue."

-- pmog's translation

I don't treat the Ultimanias as 100% canon; they were farmed out by Squaresoft to a secondary studio, and the game designers simply signed off on them. Nevertheless, I could well believe that by Yuna's day — with Sin ravaging the world non-stop for ten centuries — there were only five intact temple fayth left left plus the stolen fayth, Yojimbo.

  Pmog also paraphrases something else from the Ultimanias which I've never quite figured out:

"Fayth (other than the ones in Gagazet) were created before *and* after the Machina War (by Zanarkand and the Yevon Temples respectively). After Sin appeared, the pre-war fayth statues were then appropriated by the Temples and installed in the various temples around Spira to use for training summoners in their pilgrimage to defeat Sin."