Dog Days (anime)/WMG

Brioche's technique will enable Cinque to both remember his experiences and return to Flonyard.
She seems to know something everyone else doesn't, and she expects him to master the ability. You can't do that if you forget ever having it.
 * Jossed through the Sequel Hook ending.

Brioche's technique will turn Cinque immortal.
The same way she is; ie long-lived, once he masters it.

Flonyard is a colony of permanently Uplifted Familiars.
In the long ago past; some Familiars gained enough power in their own Linker Cores that they became able to exist without a Contract. (There are triggers that temporarily result in them taking their non-humanoid forms.) Not wanting masters, but viewed as disruptive to other worlds; they were given Flonyard. For the most part; this planet was left to its own devices, although there was one incident in the almost as long ago past where the TSAB had to step in and Administrated the hell out of them to prevent them from using conventional warfare. The natives viewed the TSAB as gods and followed the "non-fatal warfare" doctrine with such gusto that the TSAB never had to intervene again.

Leonmichelle is repeatedly invading in order to toughen up Millhiore in preparation of some imminent Big Bad or other calamity.
Supported in episode 6, where the Millhiorans suspect that monsters are becoming a problem in her area.
 * This is actually the case for the War System in general; in case the Divine Protection fails against the demons.

Cinque and the princess will end up killing each other
Now that the hero came, her vision changed: instead of one body, there are two, lying in pools of blood opposite each other.
 * Related Guess:
 * Unlikely but at least the Related Guess is Jossed.
 * Now its just Jossed in general.

Leonmichelle's vision is a Self-Fulfilling Prophecy
Tied to the WMG above,
 * Note that the palace is under protection, so it would presumably take a strong magical weapon to injure, not to mention kill, someone there.
 * While not exactly how you described it but
 * It is self-fufilling, but its obviously shown that Leonmichelle's stargazing ability is not set in stone. Screw Destiny indeed.

Brioche was a summoned hero
Her conversation with Laurent at the end of episode six, implies she's not from Biscotti originally.
 * Related Guess: Yukikaze was royalty for a now defunct kingdom and was the one who summoned Brioche.
 * Another Related Guess: Sending a hero home requires that they duel another summoned hero and survive, leading Cinque to fight, and defeat, Brioche. A possible outcome of this mirrors the next wild guess.
 * Jossed.
 * Jossed.

Cinque and the princess "die" in order for Cinque to return to his original world, and Millhiore either dies or gets transformed into a "normal" dog and lives with Cinque.

 * Jossed.

Cinque wakes up after being in a coma from hitting his head on the ground outside of his school and realizes that the entire experience in Flonyard was a dream.

 * Jossed. It happened and

There will be a crossover since Cinque is actually Takuto warped into an Non Administered World

 * Notice how the world they are in seems to have a similar style that appeared in another prominent Seven Arcs anime? And how the world produces Non-Lethal KO magic similar to that anime? How the princess of Biscotti looks a lot like some other prominent loli mage?....

Nanami will become a Summoned Hero for Gallette

 * Cinque & Nanami's friendly rivalry was a background element during the series / first season. Thus if Cinque does bring Nanami & Becky with him to Flonyard, it wouldn't be a stretch at all to see Nanami enlist with Galette while Cinque joined with Biscotti just so they could compete. That, or Nanami could simply just be a cat person.
 * This actually seems very likely.