Most MCNP runs are time independent because most particle transport is time independent. Two neutrons passing close by do not 'see' one another so one neutron only exists at any one time per thread. Two electrons would affect each other in the real world but this is not simulated.
All the tallies will produce an answer that is per source particle. More source particles simulated will make the answer more trustworthy. The statistics on the numbers will be better as they get closer to the 'true' result.
To simulate 1e18 you could enter nps 1e6, and then multiply the tally at the end by 1e18. The same answer would be true for a source twice as strong, except you would multiply by twice as much. 1e6 might be enough for many problems but too few for some.
The ps in nps does not mean 'per second' btw. I am not sure what it actually stands for but it is the number of particle 'histories' (MCNP speak for seeing what happens to a source particle) to run before it stops. It has nothing to do with the strength of the source.