Regarding the End, the trace they were trying to find is the message from a former alien from Ichika's race (probably) crashlanding on earth years ago.
She was taken back to her planet and brainwashed.

both that message we heard and the message she wrote on the tree would not change anything regarding earth's status.
After all, that first alien was on earth some years ago, probably less than 20 years ago (and overall it fits in with the Onegai Teacher/twins shows as pointed by Ryll, if I'm correct).
Those events were properly managed by the galaxy entity and no trace of those events are enough to change earth's classification and help Ichika come back.
So trying to find the place was utterly pointless from the start.

But those past events probably led to the creation of Japan's MiB...

Also the reason why Ichika comes back in a matter of weeks/months only is because those idiot aliens left Ichika's ship and Rinnon.
They trully are dumb.
With the ship alone, MiB would probably need lots of years to try and repair and then run Ichika's ship.
But with the Ship and Rinnon, of course it's a lot easier to get it repaired, operate it and get Ichika back!

Well, for the last part, I'm a little surprised a galaxy entity would change earth's class that quickly, even with some earthling coming to them...
First they should easily understand Rinnon did all the job teaching how to repair the thing and create appropriate material/tooling.
Second, a galaxy entity probably has its administrative delays... I can't see something that huge reacting under a year regardless of technology...
So the problem is not Earth's classification, but rather how quickly things got Ichika's way.