What do you mean "
its already registered"? If your wanting a dotcom, then yeah, but any other, there are plenty open.
As far as LLC goes, here in Texas, we don't "really" have to include the LLC, even though I am - there is a way around that. As with what
@Mr. Ingram's Computer Repair & Shoppe did, I too have a DBA which does not include the LLC at the end, so I don't have to include the LLC anywhere on my sight, flyers, business cards, etc. This should also apply to you, especially since your a home based business.
On another note, you can use your DBA only with the LLC as a backup for legal issues if they arise. And get business insurance. I use
Hiscox, great insurance and covered up to $1,000,000 and is really affordable. I pay just $150/yr.
I understand its NJ, but you should be able to put a logo on your website without the LLC and just put the LLC in the footer. This will satisfy their stipulation because technically, they don't say where it has to be or how many times it has to be, just that it has be listed "somewhere".
As far as the logo is concerned, its loud. Choose neutral colors and that font reminds me of techno. I went through 4 different logos before the one I have now. And the one now started out as a power button circle with a Y in it, then just a circle with a Y in it and then decided to warp it in photoshop and sectioning it off with colors. Or you can use
Fivver or any other logo design company. Or hell, better yet, buy a
logo design program. Dirt cheap and you own it and any design you make. I have the exact program and use it for for my clients when I design their websites.