Question about the setting of DFT (Density Function Theory)-NEB (Nudged Elastic Band) calculation
There is not a general rule for choosing the right displacement between images. For very big systems, I sometimes launch two NEB calculations: one with shallow parameters (eg: steps of 0.5 angstrom on the reaction coordinate), and later I use the almost converged geometries near the TS for another NEB calculation, with finer (less than 0.1 angstrom) steps. Of course the nature of the system might help you choose: sometimes things behave funny, and you have to choose a strategy for each system. Anticipating your last question, freezing or constraining sometime helps: while it doesn't improve convergence, it can speed up the calculation (by telling the program that some movements are forbidden, so that the NEB algorithm doesn't need to try them before discarding them). The intermediate image should not be mandatory for a NEB calculation (if not given, it is generated by interpolation and optimized at every cycle), but it can speed up a calculation, if provided. It can be taken, for instance, from a geometrical scan, or from an approximate TS geometry.
As for the core number, if you are talking about processors, then generally speaking as much as possibile is a good choice (edit: provided that the software you are using is capable of parallelizing efficiently the slowest step of the calculation: your choice could be the same you use for geometrical optimizations at the same level of theory)