Is there any way to transfer items out of your storage chest or off your character to the inventory of another character? I know you can do things like this in multiplayer, but I want to do it with a single player character and his items.
My space is filling up fast, and I loved how easy it was to do this with Titan's Quest and with Diablo 2(when you used the ATMA add-on in single player, that is).

Well, there is "muling": you start a co-op LAN game, enter it, drop all your to-keep items on the floor and SAVE the game. Now, exit without exporting your character. When you load up this game, either with this char or any other that can enter it (difficulty-wise), the objects will be there.

Does this affect your ability to play single player still? I know in Diablo 2, there was no going back and forth with single player and multiplayer characters except when a server got reset and bumped multiplayer characters into single player mode only.
Also, can I start a LAN game with nobody else in it but me? Or do I need to find someone to join me?

You can make a LAN game for yourself.
You can use your SP char in MP (you only have to export him/her, which you can do by clicking the appropriate button in the in-game menu). The only catch is that you can only import a char into SP when starting a new game.
This means that you can export your SP char, import him/her into the co-op MP game where there is the item you wish to give him/her, pick up the item, leave the game and start a new game with that char, but not import him/her into your already running SP game.

Ugh that sounds terrible. I don't want to restart a new game every time I want to transfer an item. :(
I guess that means I'll be facing some tough decisions on what rares, set pieces, and uniques to throw away, then.
Do you know if there's anything like Diablo 2's ATMA add-on program out there, to give you infinite and/or transferrable inventory?

There are no mods for the English version of Sacred. (Even for the German, there is only an unofficial 2.29 patch, which doesn't do anything like that, plus some new textures.)
Still, muling is good for keeping items you don't want, but don't have the heart to throw away for good.