Dijkstra for negative weights by adding a constant
No, this doesn't work. A path of length $\ell$ will have $\ell K$ added to its weight, so your transformation can make a path with several light edges weigh more than a path with a few medium-weight edges.  (Consider the effect of adding $2$ to each weight in a path of two weight-$1$ edges vs a path of a single weight-$3$ edge.)