Since I always ask questions, and almost never dissiminate information, I thought I'd add just a smidge to our available info, and invite commentary on my personal conclusions. 
The surcoat: we all know what it is. We know of the various sources that list it as keeping the armour cool in the sun, or to keep the armour clean, or to show the art of that latest craze, heraldry.
Well, I won't presume to know exactly why or exactly when it came into general use, but I thought I'd share a neat living history experience to show exactly what it DID do. This past September at a timeline event, Novae Militae was set up. The Saturday began as a hot Georgia summer day, with temps in the high 80's. Later in the day, however, it started to rain, and rain it did. I was wearing a butted mail hauberk, ungalvanized over a (painfully
) thin gambeson, and over all a sleeveless surcoat.
Here are the things I 'discovered':
1. The surcoat really does help to keep the armour cool, especially if you're going to be out in direct sunlight. To test this theory, I took it off for a while and went to stand in the sun, for about 15 or 20 minutes. The armour got quite warm to the touch, almost hot! I went back into the shade to cool down and stop sweating. When I'd aired out for a while, I put my surcoat on and went back out into the sun for the same amount of time. While I still got warm, I didn't get nearly as hot, and the armour stayed cool, cool as if it had been in the shade. So, surcoats really do help with this factor.
2. And so it began to rain. Every instinct told me to run for cover, strip off my hauberk, and save it from rust (and my elbows and back from having to clean it later). Fortunately, my dedication to living archaeology got the better of me: I left it on. And so, here I am in the rain in a hauberk. In about 10 minutes of a medium rain, my sleeves were starting to turn a little brown. The mail under the surcoat, though wet, somehow didn't start to turn brown. After about 30 minutes of this, my sleeves were really brown and the covered mail as yet was fine. At this point I saw something that made me decide to save my hauberk while I still could: another fellow had only a hauberk, NO surcoat or covering. In the same 30 minutes, his hauberk had rusted... BADLY. I mean, it was horrific! It was literally encrusted with rust, so much that you could see through the links in some places. Amazing.
3. Try not to wear armour in extreme heat or extreme wet. 
Comments? Questions? Observations?