Hanging: I recommend using a nail, screw, or hook to hang your glass piece - something sturdy. Suction cups will weaken over time and come loose from the surface they're attached to, which will result in a fall that could damage the piece.
Care: I finish all of my pieces witha protective wax polish to prevent oxidization of the solder. Over time, some oxidization can still occur, which will have the appearance of a white haze on the solder lines or near the joints of any framing. This can be gently buffed away with a soft cloth or toothbrush. If you wanted to go the extra mile, you could always apply a new coat of carnuba wax - though this shouldn't be necessary.
Safety: the solder that holds stained glass together is made of lead. Make sure to wash your hands after handling your glass piece, and not to let children or toddlers handle it (everything tends to go in their mouths and that's what we want to avoid with lead!)