Yes obviously. If you are selling anything that has your name then it must be registered. Period. As it will be designated to your name, and if you don't do it then your name will be of no use. If you were selling on any platform like I am doing it through Walmart: https://www.reecoupons.com/view/walmart I don't need to register as I am using Walmart's name.
No, It is not necessary to register your newly started company, but my suggestion is that you do that as early as possible. Like the company I have https://www.pennysaviour.com/ is not registered yet but surely going to register it soon.
Hi, this is Stella. I started Wadav.com about a year ago. But I registered it first. There are lots of business registration sites where you can register your business easily.