Thought I'd clarify a few things, just for the sake of doing so. I actually see the questions pop up quite a bit, and I figure clarification never hurts, even if it runs a tiny bit off tangent.
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Spoiler Canonically, the Grand Company leaders were in fact far from the battlefield itself, as was mentioned. They had both linkpearls, and runners to coordinate their forces on the Carteneau Flats. After Bahamut broke free of Dalamud, they did in fact give the order to retreat, as they knew any further fighting would just bring about unnecessary bloodshed. The Grand Companies were ordered to cover the retreat of the foreign brigades (I.E Adventurers), however this actually didn't fall through for any reason other than the fact that the Adventurers refused to leave. Storyline wise, adventurers who were time-skipped were all essentially on that battlefield at the behest of Louisoix. They were there in order to help defend him as he began the summoning ritual. So, when the order came for everyone to retreat, they instead converged on Louisoix in order to defend him in his last ditch effort to spare Eorzea from destruction. If you were to go by the original storyline, your character was specifically requested to be there, but there were many adventurers present. Louisoix entrusted the aid of as many capable adventurers as he could to ensure his ritual wouldn't be interrupted.
Ultimately, there's really only a few other viable reasons for a character having been sent forward. The spell itself was cast because those adventurers that Louisoix called upon were his last hope, in a sense. Should his ritual fail, they were to be the ones to carry on his legacy in the future. So, when all hope seemed lost, he used what strength he had left after the summoning and sent those adventurers around him into a rift in time to both spare them destruction because of his failure, and to have them be a beacon of hope in the future, essentially.
It definitely puts those of us who time-skipped in a sort of funky situation, were you to stick to canon. I know I do my best to downplay the extent of Merri's involvement, as it generally paints heavy involvement in the 1.0 storyline, despite being spread across numerous characters and not being something any one character experienced alone. He was just one of many adventurers who was prepared to sacrifice everything for the sake of the realm, and by no means one who destroyed the lunar transmitter or anything like that. Those are elements best left vague and unanswered, in my opinion. Unless, of course, you're with a group that wants to roleplay that specific storyline. To each their own.
Now! All that being said. The whole "memory loss" thing actually did not seem to be based on Louisoix's spell itself. While Minfilia infers this at the start, it's really the only idea she had at the time as to why no one could remember your character. Personally, I was of the mind that it was Hydaelyn's doing, as far as the storyline goes, as she is ultimately the one who allows everyone to remember you. I feel as though she purposefully chose to blank the memories of Eorzeans in an effort to allow the Warriors of Light to continue their work unhindered after their return. Cid mentions this best in a specific cutscene after he remembers your character due to his third eye. He mentions that it's best your identity remain a secret, as it would draw the eyes of the world upon you, your enemies included. So, Hydaelyn essentially gave your character the means to remain unnoticed, and once your job was finally complete as the Ascian threat thought to be dealt with, she removed the veil over everyone's eyes.
It seems like a lot of people seem to play with the whole memory loss thing differently, as most of us choose to not roleplay a character that has any involvement in the main storyline. I know for me personally, there were certain characters from Merri's background that had no recollection of him during the five years he was gone. One of the more notable characters actually played it off that her character knew she had lost some one important to them, but didn't know why or how. They just couldn't remember. However, when Merri returned, that veil was lifted, and they remembered everything. Including the fact that they had forgotten about some one who was so important to them.
Ultimately, I feel as though no one answer to the mind-blanking effect of the timeskip will please everyone. It's just a really strange concept in general. It really just boils down to going with what works for you, and the people you roleplay with.
All of that aside, during the time skip, the Grand Companies worked together to rebuild Eorzea. They were quite active, both in defending the realm against the Garlean threat, as well as the plethora of monsters that now roamed unchecked across Eorzea. They were instrumental (alongside a rebirth of the adventuring trade) in defending laborers who sought to rebuild after the calamity.
And I apologize if I came off particularly snarky, since I too was posting in a hurry from work.