How should I hang my stained glass? 
My suggestion would be to hang your stained glass on a sturdy hook that's screwed into supportive material, rather than using suction cups or sticky strips with hooks. While these options may last for a while, they could come loose causing your glass to fall, which could result in breaking or damage.

How should I clean it?
I thoroughly clean your piece and then apply stained glass finishing compound which contains carnuba wax. After it dries, I buff the glass and solder lines until clean. With this process, your glass should only need to be dusted from time-to-time with a soft cloth. 

Despite this process, sometimes oxidation of the solder may occur, which would look white/dusty. If this were to occur, you can brush the solder with a soft bristle brush. 

Maintaince:

While you don't have to do this! If you want to shine up your solder, you could consider reapplying a good car wax that contains "carnauba wax." Carnauba wax is what is in the original stained glass finishing compound. Once the wax dries to a milky, dry finish, you just need to buff your piece with a soft microfiber towel, until clean. 

Handling: 
Stained glass is glass--so it's fragile! Handle with care. 

Notice about lead:
Besides this, stained glass is held together with solder which contains lead. When lead is ingested it is toxic/harmful. When handling glass, take precautions such as not touching your face or mouth, including eating and drinking before washing your hands. Washing your hands with warm, soapy water thoroughly after handling.