It seems the only issue is, that you have incredible high latency to one another.
Of course, here goes! (This is using Evolve)
See the issue? Evolve will make the game traffic go through additional hops and might slow the connection down generally (the same as Hamachi) and increase the latency ...

His Log was lost, but we started a new game with new logs and pictures. This time we loaded without mods (but still same problems). I haven't had any problems with connecting to others, but my brother seems to have the same issue wherever he goes.

My log (Host)
0.002 2017-02-06 00:13:37; Factorio ...

Of course, here goes! (This is using Evolve)

0.002 2017-02-05 22:16:07; Factorio 0.14.22 (build 25342, win64, steam)
0.002 Operating system: Windows 10 (version 1607)
0.002 Program arguments: "C:\Program Files (x86)\Steam\steamapps\common\Factorio\bin\x64\Factorio.exe" "--wait-to-close" "7248 ...

Me and my brother have tried to play with eachother for a couple of days now, and no matter what we do we still keep getting the same results. When either one of us makes a server, the other person keeps getting constant "Losing connection" popups every 5 seconds or so. We live quite close to ...