Is this bell circuit safe

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!

The configuration you pictured is not really a good idea. Waste water from the washer is going to flood the heater drain pipe, leaving nasty residues that could conceivably siphon back into the heater if connected to the heater drain valve. Not likely, but possible any time two drains are connected at different levels other than directly into the vented out flow pipe. The other problem will be that if the drain is a open pan drain only, washer waste water will back up into the pan. Washers tend to eject water quickly and can partially fill the pipe above the trap.

I would rather see you pump it up to the inlet of the washer drain.