Moss prefers moisture, shade, and acidic soil conditions. One thing that may be happening in your yard is that as trees get older, they create more shade, and that in turn can keep the yard from drying as quickly. You might want to consider thinning some of the branches on your trees if this is a possibility. Letting in more sunlight will help prevent moss from growing in your flower beds. Sometimes, it's also a matter of correcting drainage problems to get the soil to dry out faster.
There is no particular connection between moss and grubs, although any plant under stress is more likely to be attacked by insects.