I viewed your website when you had the other background and I think it looked a bit better than just a plain colour, as I thought the darker colours made it look better in my oppinion.
As for the logo, just design something on a transparent background, that way whenever you change the background to the website the logo will just sit effortlessly on top of it. Perhaps just by using a funky font on the image, and having the word cCorE on it, would make it look nicer than just plain text .... then again imo anything looks better than plain text for a logo.