Where I live, you have to register any business that is located in that city. Even though you are doing your business online your physical location/office/where you work is in your home in your city. You shouldn t have to pay taxes to your city/state for items sold out of state but if it is sold in your own state I believe you legally do have to pay taxes on that item.
No. Everyone sells on ebay. Registering would only be necessary if it is something you might do long term and earn a steady income from. Where you might have to pay taxes, etc. Second hand items on ebay won t matter unless you are making gobs of money, use a business name, etc.
No, you are not conducting business within your own city. You only register (but really apply for a liscence) if you have a brick and morter location. Though you still should report the income on your taxes, depends on the signifigance.