This is interesting. First, you would actually have to trademark your name. (see tldr below). As for the streams and videos: there are lots of sites out there that will file copyright forms for you, but most are geared towards music, film and music videos. If you treat your broadcast and any videos you sell like films, then any one of those sites should work to register the videos for you. But, you must have the recorded work to submit, so you would need to record your streams and submit and register them as individual works.
:tldr:
In the US and most countries, copyright ownership by the creator is inferred when the work is in tangible form. You don't have to register, but registration offers added protections and recommended in case of lawsuits or counter-claims of ownership of your work. To be be absolutely covered under US law, you would have to record your stream and register and upload a copy with the Copyright Office. If you follow copyright infringement cases (mostly in songwriting) the burden of proof has been on the creator to prove that they actually created the copyrighted work.
A lot of these sites sharing recordings of shows fall out of US law, operate in countries with shady copyright laws like Russia and China, and with enough heat will just shut down and move everything to a new site. So, registration and DMCA claims might really not do much with the shadier sites. The "legit" sites like PornHub are very much worried about copyright and will
The DMCA does a bunch to true up copyright law internationally, but, there are still flaws and holes in it.
As for your name, names and titles cannot be copyrighted. Business names, however, can be trademarked. Unlike copyright, in the US first use does not always guarantee that you will be granted ownership of the trademark in your name. You need to be using the name nationwide (meaning at least across 2 state lines), and no one else is using it nationwide. This is why you can see so many unrelated businesses with the same name in different states, but only one Hot Topic, Macy's, etc. There is a band called Suede in the UK that formed in the early 80s. They had been releasing albums and touring throughout Europe for years. Then a band in the US formed in the later 80s with the same name and only performed in Texas, Oklahoma and Arizona. Suede from the US were granted ownership of the trademark because they used it first in the US. So, if "SensualJade" was being used for e-commerce by someone to sell whatever sensual product, you might not get the trademark.
This is not a substitute for professional legal advice. I am passing along my experience registering and protecting my works and my former employer's works as a songwriter, video producer, software creator, and former employee of one of the largest international cable networks.