You are trusting someone else your funds. if this doesn't scare you enough, then I don't know what scares you... There are simple ways to have offline storage (search for air gapped computer, offline storage and Electrum)
So, what I am wondering, and hoping someone can further explain to me, what are the basic risks to leaving any relatively significant amount of coin in an online wallet, such as GreenAddress, for any long period of time? Is the risk someone hacking into their servers and gaining access to your coins? Does this multi-party address resolve that concern? Is it the risk of GreenAddress having fraudulent internal compromises? Is it the risk of them no-longer being around in the future? I totally understand that the safest way to hold coin is to have a "paper wallet", but what are the specific... or what is the most likely way that an online wallet would be stolen, my private key exposed?
These are exactly the issues you're facing: hacking.
I don't use GreenAddress, but how's their multisig security? do you have to input your key always to spend your funds?
I think you can be MITM attacked... So even if they have only one part of the key, when you input yours you can be hacked.