When I change the uniform and look of a Starfleet bridge officer, I have the option of giving them the All Good Things/Visitor/Timeless/Endgame alternate future commbadge. I'm sure at one point I was able to give it to my first ever player controlled character/captain as well. But now, when I try to change the commbadge for one of my newer captains (who I made a few days ago), I can't find the commbadge for her, but I can for her bridge officers. Is the commbadge only available for certain Starfleet uniforms?
It should be available across all uniforms, or it certainly has been for the majority (if not all) of mine. I tend to have it on all of my characters now because it just plain rocks! It was a Veteran reward though, I believe, so it should come up immediately on log-in with your new character. If it's not available at creation, it should be when you visit the Tailor.
Not sure about this, since I don't have any alts, but you may have to claim your bonuses for each character. Make a new character, hit the store again to claim your bonus.
It's not a C-Store item, it's part of the 100 Day Vet reward and is account wide.
You should be able to access this option after earning the 100 Day Vet reward. Once you have created a new character and then logged into the game, that is. Kind of a pain but there it is. All existing characters should simply log in and then hit the tailor.
It sounds to me as if you are experiencing a bug since you had access to it and then lost it. All I can recommend is to post an in-game ticket / bug-report and hope they fix it. To answer your specific question, yes, it should be available with all Starfleet uniform options.