Just to get things started, here's a proof for #1:

Proof by induction that the number of bicolor edges is odd iff the ends don't match. Base case: a single node has matching ends and an even number (zero) of bicolor edges. Extending with a non-bicolor edge changes neither condition, and extending with a bicolor edge changes both; in both cases the induction hypothesis is preserved.

Just to get things started, here's a proof for #1:

Proof by induction that the number of bicolor edges is odd iff the ends don't match. Base case: a single node has matching ends and an even number (zero) of bicolor edges. Extending with a non-bicolor edge changes neither condition, and extending with a bicolor edge changes both; in both cases the induction hypothesis is preserved.