If the battery is dead after disconnecting, it is probably worn out and no longer holding a charge. However...
The "blinking" means DO NOT DISCONNECT. While the light is blinking, the light will never turn green, even if the battery is charged.
The reason it may be blinking all the time... Select the shuffle in iTunes, and go to its Settings tab. If Enable disk use is checked there, iTunes keeps the shuffle's "disk" mounted continously, while the shuffle is connected. Therefore, you need to eject the shuffle (unmount the disk), before physically disconnecting it. After you click the eject button in iTunes, but before you disconnect it, the lights should stop blinking; it's now OK TO DISCONNECT, and if the battery is charged, it should be green.
If Enable disk use is NOT checked, iTunes only mounts the disk when syncing. The rest of the time, the shuffle's disk is not mounted, so you can disconnect it without first ejecting it. Therefore, the light should not be blinking, and it will go from orange to green when the battery is charged.
For iTunes not opening automatically when the shuffle is connected, check the setting on the Settings tab that says Open iTunes when this iPod is attached. Make sure it is checked.
For the battery not holding a charge, if it persists, you can try doing a Restore on the shuffle (from the shuffle's Settings tab in iTunes). This will erase the shuffle, re-install its software, and set it to default settings. If a software glitch is causing the problem, a Restore should fix it. But in this case, especially with a 1st gen shuffle, the problem may be the battery itself.
