What is the definition of the validity of JOCS, exactly one cell can be determined or what else?
i.e., is it possible in the Competition Puzzle that all other puzzles are completed, and (0 or more) cells are deducible in JOCS (and these are not part of the overall solution)?

This is same as the JOCS that appeared in WSC 2010. Exactly one cell can be determined.
Since it has connectors, identifying and filling in the connecting digits must be done before solving it as JOCS.

The connecting digits can be filled in without solving the JOCS.
The JOCS will not be solvable until the connecting digits are filled in.

There is only one solution to the JOCS after filling in the connecting digits and there is only one overall solution to the whole set of sudokus.

In Half-Squares Sudoku, when a connection is made to a half-squared cell can the digit be on either side of the diagonal line or must the digit be on the side of the diagonal that the connection touches?

There will be no connectors on a half-squared cell :-)