SpoilerCrystalline as a linkshell was originally something that Eva's father had had created a few decades ago when he was living in Ishgard and maintained ownership of throughout the years.  This is something that he and a few [now more elderly] characters had used back in their "glory days" to keep in communication with each other.  It was never any sort of formalized company or anything more than a group of friends who leaned upon one another during some tough times of their own that the previous generation faced.

Throughout the course of 1.0 pearls were somewhat randomly dispensed by Eva to those she considered to be friends.  She had been a part of a number of communities, some of which were a bad fit for her but she wanted to keep in contact with friends she had made, and other companies didn't stand the test of time for whatever reason.  Towards the end of 1.0 she was heavily invested in a group called Everwatch as a sort of assistant to its leader Oskar.  When he was feared dead after particular events the company disbanded, but more of these pearls were distributed to the members in order to preserve friendships and the bonds of camradarie.  At that time there was little sense to trying to breathe new life into a company because Dalamud was swollen in the sky and people were panicking.  So the little blue pearl that her father had given her before passing away wound up being handed out as a way for people to hear the comfort of their friends' voices during what looked to be tough times.

And then the Calamity happened and all the pearls stopped working.  Eva herself was pulled into this time rift along with a number of other members.  Some members remained behind though and survived throughout the five years.  They have their own stories and those characters have seen marked changes as well, from what I have come to understand.

Now five years later the people that were safely removed from that battlefield are beginning to re-appear in Eorzea - confused, disoriented.  They're finding that their homes have been destroyed, or reappropriated, and that life has gone on for five years without them.  And for many of our characters that's been a tough pill to swallow.  They're meeting with slightly-older versions of characters they once knew, but are finding that they're not being recognized.  And amidst all of this, some of the older "original" elder members are making their presense known as well.

So we have a lot of interesting sort of interactions now where Eva's still adapting to the change in environment, adjusting to people she knew from only a month or two ago [from her perspective] exhibiting five years worth of change and in many cases having to convince them she is who she claims to be since there is that fuzziness of memory being exhibited by many - and it's not always quick for that realization/recognition to sort of 'click'.  And some newer folks have been rolled in as friends of friends and such.  And the old timers are kind of back offering their advice and sort of gently trying to nudge Eva along the path they think her father might have wanted.  And some who shared the time jump experience are just finding solace in each others' experience, or trying to help find other friends who may have gone missing, or dealing with that uncertainty of not knowing whether their friends, lovers, or family are alive or dead.

Somehow this will give way to an actual company forming with Eva (not a natural leader herself) at the helm somehow, but for right now there's just too many really wonderful sorts of interactions to explore first and it would be a shame to kind of rush through all of this chaos for the sake of hastening the company's formation.

Hello, main scenario quest spoiler!
Because you must have the Echo to not be tempered by a primal, and your first primal fight happens with Ifrit around level 20-22, it is more or less impossible for a PC to continue doing primal content IC if they don't have the Echo (since you'd be tempered by Ifrit and immediately become an enemy of the Scions, the Empire, and any adventurer allied with either side).

Additionally, we know from the ARR quests for legacy characters that having the Echo makes you immune to the amnesia regarding the Warriors of Light. In fact, Minfilia states that explicitly when you first meet her in the first Scions plot arc (around level 17).

So, if all PCs in your FC have the Echo -- which is required to do content IC past level 20ish and certainly at endgame, not to mention being part of the main scenario quest -- they would remember you (if they'd met you before, at any rate), and as such, there'd be no issue.