I have a 2006 Honda Civic EX and experienced the same issue under similar circumstances and have since solved the issue:
1) Cold weather 2) Car was recently jump started 3) Receiving flashing green key indicator 4) Car starts then shuts off 5) Battery problems
I went outside one cold winter morning and the car was dead. Click click click. A friend of mine came by to try giving the car a jump, which allowed me to get the car to start, I could even rev the engine once or twice, but then, like something told it to, the car would die, all the while flashing a green key on on the dash display (This green key is in fact the indicator for the cars immobilizer system. When you start the car with the key that matches your car, the green key will flash solid once, and the car will start fine. If the key does not match your car, the green key will flash multiple times, at which point the car will die).
I continued to reproduce the same results previously described, but eventually the battery would no longer recharge from just the power of my friends small sedan.
Ultimately, after being towed to my regular mechanic, and then the dealer, the stated problem was that one of the cells in the battery had gone bad, and I just needed a new one. Looking closer at the bill from the dealer, they did in fact have to first reprogram my key to the cars computer. The issue started with the immobilizer. Apparently at some point, one of the two was scrambled, and my car was no longer accepting the key. After a number of tries to jump the car in cold weather, with a bad battery, the immobilizer issue no longer mattered, because the car's battery no longer had enough juice left to allow for anything but some clicking, let alone starting and activating the immobilizer.
Short story long, I did a little digging and what seems to cause the key and the car to no longer communicate properly (which activates the immobilizer) is jump starting the car with the key in the ignition. This should be a no brainer to begin with, but it happens, obviously. Don't have your keys in the ignition when jump starting your car. It is possible that the key can be in the first of the three ignition positions without causing the problem, but I'm not going to try and find out.
Fortunately, the dealer (and some competent honda mechanics) can quickly reprogram your key to the cars immobilizer unit (literally found right under the ignition). Unfortunately, the issue is typically paired with a battery problem, and if you can't get the car to start, it needs to be towed in order to be reprogrammed.
If you're experiencing this problem:
1) Bypass your regular mechanic and have the car towed to a Honda dealer. Your car's immobilizer needs to be reprogrammed. Tell them the circumstances leading up to the problem, including anything involving recently jump starting the vehicle, the green flashing key light, etc.
2) Inform them of Technical Service Bulletin (TSB) 09012005
Make : HONDA Model : CIVIC Year : 2006
Manufacturer : HONDA (AMERICAN HONDA MOTOR CO.)
Service Bulletin Num : 09012005
Date of Bulletin: SEP 01, 2005
NHTSA Item Number: 10018480
Component: ELECTRICAL SYSTEM
NEW IMMOBILIZER SYSTEM ON '06 CIVICS.
3) Make it clear you want the immobilizer checked first. Don't let them tell you you need to buy a new key.
4) Even with all of the information out there, this seems to be a puzzle to some dealers, so don't be afraid to call every day to check on the status (most likely, they will be calling you first). If days have gone by, and they have not tried reprogramming the immobilizer unit, something is fishy, and it is time to insist.
5) Wait for the phone call back from the dealer telling you either it was simply the immobilizer issue, or that you need a new battery. In the latter case, don't expect to necessarily hear/see anything about the immobilizer issue until you get the bill.
6) When you get the car back, and you start it for the first time, your radio will ask you to enter a code before it will allow any audio to be played. If you have lost the card that has this code on it that should have come with the car in the first place, check for a white sticker on the body of the glove box, normally outer left side. It will most likely be a 5 digit number next to a longer serial number. Use the displays buttons to enter the code. If you can't find this sticker, head to the Honda Navicode Website to retrieve it: https://radio-navicode.honda.com/.
7) If, like the original poster, you have had a remote starter installed in your vehicle, the mechanic who installed it would have had to put an "immobilizer bypass" unit in the car, in order for the remote starter to work. As these bypass units are doing just that, bypassing the immobilizer, they are typically not the issue in this circumstance. In fact, the car can often be started with the remote start and not the key, specifically because the remote starter bypasses the immobilizer all-together. Either way, be sure to let whomever is working on the car know about the remote starter, just to be safe.