No, it doesn't come with a cup. I use 12oz disposable cups for my coffee pods. You can set the machine to dispense 3 different levels of the drink by adjusting the cup size before you press the start button. I would suggest you test it out so it dispenses the right strength for you.
When you put in the included “cleaning” disc into the machine it will dispense hot water
#6 Does this machine operate by design with and without a water filter? (Can you just use tap water only).
Simple answer is yes! I have a water softener so don’t use filter. If you have hard water suggest using filter or will be forever descaling your machine.