1. Is there a way to install mods manually? I can't login in game to the mod library for some reason.
Yes. Currently the mods available in game are maps. At the top of this page, there should be a mods link which should give you access to the mods library. Installing the various mods should be fairly easy and most have instructions on how to install them. Most of the mods are zipped and contain folders which should be placed directly into your /my games/elemental/mods folder. Mods which do not contain such a folder and simply contain xml files should be placed into /my games/elemental/mods/data. New map files should be placed in the folder /my games/elemental/maps.
2. Whenever I make a custom faction, no matter what bonuses I give it, it is able to train Experienced units even though it doesn't have the Training technology. When I check what technologies it has completed, it doesn't have the Training technology, yet it is able to train Experienced units. Is this a bug?
This sounds like a bug. You should probably post in the support forum and see what they have to say.
