One method to detect negative cycles can be to traverse the graph to find all the cycles and then add the edge weights in the cycles to see if any are negative. Will this method be faster than the Bellman Ford's algorithm $?$ NOTE: You only need to find whether there is a negative cycle or not. You do not need to find the shortest paths.

Given a graph that does not have negative cycles, but has some negative weighted edges, can we increase weights of all the edges by a certain amount so that they are all positive and then apply Dijkstra's algorithm? For example, if the lowest weight is $-5,$ we can increase all weights by $6$ to make them positive.