Frank has given you some solid advice in regards to domains, however I'm a little more liberal in my thinking. Dot coms are of course the most widely used and preferred, but I personally like .CO as well.
Dot TV is a killer domain for anything video related. I've use that in the past and found it to be easy to market. Domains like, .info and so on, stay well clear of. They are not seen as a serious domain and won't give you any value whatsoever in building an online brand.
It also comes down to where your market is and how you are marketing to them really.
In regards to picking your business name. I wouldn't be too concerned about linking your niche product to your business name, unless that is your speciality. Linking your name to one product can serve to focus customer's attention on that product, at the expense of marketing other products where other products are of course involved.
Don't worry about ensuring your domain matches your eBay name, it's probably better that you don't if you are developing your own site as well. Using eBay as a sales funnel for your own site can be very beneficial, however eBay should only ever be used as a sales funnel, not to be then marketed in reverse by directing traffic from your other site/s back to eBay.
If you have your own site then you are not having to pay eBay fees, so the last thing you want is people using your eBay store when you already have them using your site where your margins are greater.
I would suggest using a completely different eBay name to any domain you register, so people can't make the link between your site and your eBay site personally.
Finally, no, you do not need to be a registered business to have an eBay shop or your own website.