Replacement Digital Camera Battery 3.7V 680mAh for Nikon S-550 S-560 S-660

Not if you are using a genuine battery and an original charger. If you are using a third party "compatible" battery and/or third party charger - anything can happen. Both original camera batteries and chargers have circuitry inside to prevent overcharging. When the charger senses that battery is full, it switches to trickle charging mode, when battery is getting just a little amount of charge routed to it, which helps it stay fully charged and ready to go, but won't harm the battery... So it is perfectly safe to keep the battery in charger for as long as you wish. Many photographers have an extra battery that always sits in a charger, when battery in camera needs replacement, they just switch them.

A poorly made battery may fail to properly communicate to the charger that it's full, and a cheap non-genuine charger may not be able to interpret the message from the battery (so to speak), and again, overcharge...

This is why it's important to use genuine batteries and chargers, there's a reason why they cost more - of course you're paying a little "brand name" premium, but you're also paying for the electronics inside. If your battery and charger were original ones - then you may have a case for warranty service of either camera, battery or charger, because this should not have happened!

Sorry, cameras are not like ipods that can be charged by USB connection. There might be one or two that can do this, but generally most cannot. So, you'll need to buy a replacement charger.
Check your User Manual to see what it says about charging your battery....

Some batteries have a memory; so if it is only partly depleted then recharged it will be less efficient over time. Lithium batteries are not affected by memory effect however.

So to answer your question how do I "discharge it for one complete cycle before use": Charge it fully then run the camera until the battery is completely flat i.e. the camera won't turn on. Then charge as normal.