Second, its a question about BFS (not much an AI topic itself), theres any way to keep track of the steps without storing it in each node added to the open list, but on the algorithm itself?
BFS deals with planning, and so is very much AI, even in the general sense.
With BFS you're going to have to store at the very least the parent of each state you consider in your search. If BFS is too memory-intensive for your application, consider DFS with iterative deepening. Like BFS, this is an uninformed search that also finds the optimal path on an unweighted graph. If implemented with a stack, you would only need to remember a very few states, but that algorithm does require a bit more processor time than BFS. That's often the trade-off when you optimize for low memory usage.
(Note: If you're ascribing different weights to different steps via an influence map, you're not actually implementing a BFS.)