I wanted to add a series of quick solution sketches to my problems without one. These aren't technically full editorials, but I hope that they are enough for you if you are stuck. :-)

The first subtask was meant to reward dynamic programming approaches.

The second subtask was solvable with simple matrix exponentiation across a tropical semiring.

For the final subtask, if $G$ is the matrix representing the graph, we precompute $G^2,G^4,\dots ,G^{2^p}$ and multiply an initial vector state (which represents the state after taking zero edges from the starting node) by the matrices to compute $G^{k}v$ instead of finding $G^k$ directly.