When you fold this cube net, you would get this connection:
Start from the obvious one, the two adjacent black cells:
Then some more:
What if you connect the lines below the white cells, in the 6-celled side? (shown in dark green)
So the left one needs to be connected on the black cell.
Now for the question mark:
[EDITED to add:] Oops, this is not actually an answer to the question as posed because I failed to notice the requirement that there be a unique solution. I have now verified that my surely-non-optimal answer to Q1 is in fact a uniquely solvable puzzle. I haven't given any thought to that question in regard to Q2 yet. It might be rather hard.
A surely non-...
The Russian desman
After much staring and thinking and dictionary-trawling I remain confused by 8d, which I can't make any sense of either as a p.d. or as an ordinary clue. [EDITED to add: M Oehm figured it out; see below.]
Credit where due: