Heck, in that case, you probably meet code with the existing cable if you just secure the bench to the wall(s) and/or the floor -- anything that makes the bench actually attached to the structure of the building. [...]
"Capable of being removed or exposed without damaging the building structure or finish or not permanently closed in by the structure or finish of the building." [2008 NEC, Article 100]
Not if the cabinet is permanently installed. OK if the cabinet can be moved out of the way without damaging anything.
Yes to both.
No. The master bath in my house has a junction box concealed behind a large mirror -- but the mirror is in a channel that permits sliding it aside. That box is "accessible".
The Code has another term, "readily accessible", which is much more restrictive. The gist of it is that if in order to get to something you have to move anything out of the way, or fetch a ladder or a stool, then it isn't "readily accessible". Breaker panels, fuse boxes, service disconnects, etc. are required to be "readily accessible". Junction boxes need only be "accessible".