Ipod shows its connected when not plugged in

Try a manual reset by holding the center button and the play button together for at least 10-15 seconds, this should reset your iPod and should refresh the battery,
In future do not let your battery run flat completely always keep a charge, and lastly charge through a wall charger and not from the usb port,
good luck

plug it back in

There are two 5th gen iPods, A1136 model is the older 2005 model, not the later 2006 model.

Here is Apples link: http://support.apple.com/kb/HT1353#ipodfifth2 which will show the differences, but not by model number. The reason I know this model number is the first 5th gen 2005 is because I have one. Also older 2005 should have '2005' engraved on rear.

it's a broken battery, your dock station probably charges through the firewire port and is strong enough to power up the ipod without a charged battery, the USB port alone isn't. if you're tech savy or have a tech savy friend you can use a ipod repair guide that will show you how to replace the battery step by step, or you can go to the ipod repair nyc.

This is either a bad battery or bad battery connector Hope it just Battery.

Don't just change the hard drive,Change the motorboard as well.Different generation ipods use different boards and the hard drive from one won't always just plug in and work.

It could be something wrong with the main board or a faulty replacement battery.
hope this helps at all.
good luck :)
Jaxon

No there is no heat associated with this rainbow affect.

To reset an iPod with a Click Wheel, including:

• iPod nano (3rd generation)
• iPod classic
• iPod (5th generation)
• iPod nano (2nd generation)
• iPod nano
• iPod with color display (iPod photo)
• iPod (Click Wheel)
• iPod mini
• iPod mini (2nd generation)

1. Toggle the Hold switch on and off. (Slide it to Hold, then turn it off again.)
   
2. Press and hold the Menu and Center (Select) buttons simultaneously until the Apple logo appears, about 6 to 8 seconds. You may need to repeat this step.

Tip: If you are having difficulty resetting your iPod, set it on a flat surface. Make sure the finger pressing the Select button is not touching any part of the click wheel. Also make sure that you are pressing the Menu button toward the outside of the click wheel, and not near the center.
If the above steps did not work, try connecting iPod to a power adapter and plug the power adapter into an electrical outlet, or connect iPod to your computer. Make sure the computer is turned on and isn't set to go to sleep.

probably, when you replaced the battery, you didn't properly inserted its contact to the board connector, so just open the ipod again and re-install it, make sure it's all the way down into connector