Yes. It is known as self-discharge and all batteries have it. Some batteries have a shelf life measured in years - lithium primary (non-rechargeable) batteries and alkaline batteries have an estimated 10 year shelf life. Rechargeable batteries are less, typically a few months to a year.

Yes all batteries would inevitably loose power over time if the car isn't started, because then the alternator wouldn't get a chance to re-charge the battery. Usually if you don't drive a car that often or if you are away for extended periods of time, people buy a "battery tender" which plugs into a house outlet and keeps your car battery charged.

Invest in a Battery Tender Maintainer if the car isn't driven at least 20 miles weekly. The older the battery the faster it discharges.

Yes they lose charge over time. More so if it's connected to the car since the car uses the battery to keep memory on it's electronics like, radio, ecu setting, clock, alarm.

Yes it will. Some will take longer than others, but they will lose their charge.

Nope it will have power for years and years and years if you don't use it. yep, years and years.....