No, more frequent use than once every couple months is recommended, to keep your battery healthy. Running on battery for a few hours every few days is recommended. See the link below for more information.
Yes, on all Apple notebooks, when the battery is fully charged, it stops charging, even if left plugged in, and runs on AC power. Only in rare circumstances will your MBP draw power from both AC and battery. You really don't have to worry about your battery.
Run on battery whenever you need to and plug it in whenever you can. You can plug or unplug any time you need to, regardless of the charged percentage, and you never need to completely drain your battery. Just make sure you don't run on AC power exclusively, as your battery needs to be used regularly to stay healthy.
The link below should answer most, if not all, of your battery/charging questions. If you haven't already done so, I highly recommend you take the time to read it.