Partial answer: solved the Masyu puzzle. (My first time solving one of these!)
The letters not covered by the path spell:
Argh! It seems this puzzle just got bigger. Here's the new version:
Solution to that one:
This time the unused letters spell out
First of all, let's adopt the convention that the path runs "macroscopically" anticlockwise. Now,
Now, in a Total Masyu puzzle, when the path enters an empty square
The path forms a single loop, so
In that case
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:
There are two solutions for 7x7 grids
(not counting reflections and rotations)
I feel a bit guilty posting this as an answer since I know that at least StephenTG had done most of it before
and also because frankly Rand did most of the work. But here we go. After Rand's solving we have
[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-...
Partial answer - stuck on extraction
Solution to the Nurikabe:
What else is going on?
I'm not sure what to do next; overlaying the solutions to the logic puzzles doesn't look like anything useful. (And the fact that the 4×4 doesn't appear to work is surprising...)
With thanks to @Deusovi for solving the Masyus and doing the hard part.
If you overlay the Masyu solutions on the Nurikabe solution, you will find something interesting:
You notice that
So to make this Masyu more colorful, you should probably do this:
There, much better.