I dont know if there are two seperate problems here...
If there is no picture when direct to TV.. it could be a number of things..
But plugging the nextbox 3 into the DVD recorder, by any other means than just the composite video (regular single RCA video), likely will not work.. the NB's output copy protection. I know its enabled on the HDMI, and even on the component... though you can find it present on composite as well ocasionally (ran into that in the past with trying to copy VCR to VCR with a bought VHS tape.
The 8300 has the option of recording to VCR (or DVDR). The Nextbox 3 does not have that option. Whether this is by design or oversight we don't know. In any case, the new NB3 has a lot of other bigger problems that need fixing, so don't expect this update anytime soon.