Regarding the relation between population size and exploration or exploitation. I would say that the population size influences proportionally the exploration. The more chromosomes the more solutions are explored each generation. Additionally to the explanation above by Ales, the elitism size influences exploitation, on the other hand, mutation rate influences exploration. From my point of view, the crossover rate influences both exploration and exploitation, because it creates new chromosomes (exploration), however, these new children chromosomes are not created out of the blue but from parent chromosomes from previous generations (exploitation).
I want to look about the relationship between the mutation and crossover ratios with the size of the population, is the size of the population is directly proportional to the ratio of crossover and inversely with the rate of mutation or vice versa.
More population means larger initial diversity of algorithm & hence larger parts of the search space can be covered per iteration. This will increase the exploration capability of the genetic algorithm.