Electrical – live and neutral, led driver

Key difference:

• Halogen lights generally run from AC so this is what your transformer outputs.

• LED's require DC so generally yes, you will need a rectified output from your transformer.

I am guessing the transformer is providing enough forward voltage on initial switch-on, but not thereafter.

You may be able to add a rectifier circuit at each bulb location to cope with this.

You won't be able to shock yourself with 12v, but you can shock yourself with whatever power supply you find that'll do 5A at 12v. Assuming you'll only use one strip at 6a just grab a laptop power supply (I'd recommend this one: http://www.newegg.com/Product/Product.aspx?Item=N82E16812203037) - that way you really don't need to worry about whatever engineering decisions went into building that laptop power supply.

If you cleanly splice the wires, you won't have any issues. Solder them correctly, twisting them to have a mechanical connection and soldering to have an electrical connection.. then heat shrink it.

The LEDs will not give off enough heat to catch on fire or burn your bed, ever.

This project could easily be the fusion of 2 well engineered products to do something cool without any risks! Have fun!