LiPo's are safe, you just need to use a proper charger and make sure not to over-discharge. I work with them at work and have a programmable charger that cost $100 but is very nice. You can get cheaper chargers that are OK for $30 or so. I think Groupner makes one.
Never charge to more than 4.2V, and never go below 3.0V. They have a very flat discharge curve, staying around 3.7 for most of their charge life.
They are neat little things, right now I'm working on a project with a 150mAH single cell that will do 10C continuous discharge. Nifty.
Oh, and I have no idea what that board is. Might be protection circuitry, but who knows. I would remove it. Make your own protection ckt so you are sure what it does, if you feel you need one. Also, does the cell pack seem puffed out at all to you? Or is it flat?
Oh, and don't connect the LED straight to the leads. That could be very bad. If it's forward voltage is much less than the battery voltage, you will dump a LOT of current through the LED and it would be bad for the battery.