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.

If we imagine blue as labelling the odd numbered segments and green as labelling the even numbered segments, it is clear that there must be an even number of segments in total. The number of gaps between segments is equal to the number of segments minus 1, so it is odd.

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.

Here's a more conceptual framing:

If we imagine blue as labelling the odd numbered segments and green as labelling the even numbered segments, it is clear that there must be an even number of segments in total. The number of gaps between segments is equal to the number of segments minus 1, so it is odd.