From the manga: after the battle at Agharta, Mikako sees Hisa and Noboru's names on the crew roster for the relief fleet. Since the FTL drive circumvents relativistic time dilation, this means that she's seeing the roster during the earth year 2048 (or somewhere around there). Noboru was on earth when he received Mikako's message, meaning that the his fleet doesn't launch until at least around 2056. Further, since the Lysithea is limited to speed-of-light communications, it means that the message would have had to have been sent around the earth year 2040, 7 years before the Lysithea launched. (Dates are rough approximations)
As manga wasn't done by Shinkai himself, IIRC, the most probable explanation would be that mangaka simply screwed up with the dates. It's highly doubtful that Shinkai, who's known for his painstaking attention to the details (and scientific accuracy of his SF works), would do that major a mistake.
No, the message was sent in September 2047. Remember that the message is one-way. It would take 17 years for Mikako to receive Noboru's reply, but only 8.6 years for Noboru to receive the original message.
Why isn't is standard procedure to send small ships back and forth between the UN fleet(s) and Earth, carrying messages? FTL travel can't be too expensive or dangerous of itself, because the fleet went to warp unprepared, and jumped to a random point one light year away from earth, without everything going to hell, and it can't be dangerous because of Tarsians because the jump is precise enough to safely end up near planets. So if you just have one of the minor ships serve as a messenger between earth and the fleet, they could ask for reinforcements or repairs and help reduce text message travel time from years to somewhere between hours and months (depending on the cost of FTL). And then, Earth would know something was up when no message ship arrives, and send help in hours-months instead of years.
And more importantly, it would improve morale among the troops. Most of them have loved ones back on Earth whom they can't talk to and who do not know whether or not they are still alive. That has to be brutal on everybody, to the point of impairing operations.
A possible explanation (not saying that's the explanation, just that some sci-fi settings do use this plot device, so it might be the case) could be FTL drives being large and cumbersome, requiring big ships, meaning "no small FTL-capable courier ships for you".
The manga does have severely-wounded personnel getting sent home, for what it's worth.
And another thing, how long is Mikako's tour of duty supposed to be? The fleet she was on was doing 'reconnaissance', and she's only been gone for a few months in her time (dates of text messages: July 3rd from Jupiter, August 4th from the one-year waystation, September 16th from Sirius B, all in 2047). If it wasn't for the fleet being attacked, why wouldn't the fleet just return to earth once every year/once every solar system, and so be back on earth before the first text message from outside the solar system even arrived? Mikako seems very certain she won't see Noboru for eight years, after having been gone for only a few months.
The most likely reason could be that the time they are living in is radically different. By the time Mikako sends a message eight years have passed in Noburu's lifetime, she might seem so despondent since she might have a feeling that the distance between them and time of communication will only continue to get longer. It seems that the fleets just keeps on going as far as they can. She might might even think she'll face a day when by the time a message reaches, his lifetime may have already passed. If you consider the anime's time period. -If the manga is something to go by, she might get happier news soon or at least while they're still living.
Why is the UN Army drafting 15 year old girls?
Probably so they'll be old enough to keep fighting in the military/training others when/if they return to Earth.
How does their FTL drive control when they emerge from the wormhole? Two events separated by a spacelike interval don't have a defined time interval between them, nor even a defined sequence. A 7-light year jump could result in them emerging up to seven years before or after they entered the wormhole.
There was a different sci-fi story (The Light of Other Days, I think it's called), where the wormhole generators made their 'paths' in space, with zero or little relative time difference. Then it was discovered how to tune the generators to make the 'path' in time, with zero relative spacial difference. Maybe the FTL drives here are only tuned to the 'spacial path' because they haven't developed them further (or the physics won't support it — it apparently supports RoboTech-ing instead).
Why is it the UN Army? They operate in space!
They could operate on a unified joint structure like the Canadian Armed Forces. So "Army" could be a catchall term rather than carrying the specific connotation of the land arm.