Coniferous plants can be very difficult to identify. It looks like a Chamaecyparis, possibly Chamaecyparis pisifera. Without a photo of the cones, this is a guess at best. Shade and root competition kill grass under trees. You might consider planting a shade loving drought tolerant ground cover instead of grass under the tree. vw
Actually, in the 8 years I've been here, this tree has NEVER produced any cones, or flowers for that matter. Yes, I do understand how plants/trees reproduce. All this thing ever does is shed it's leaves, little bits at a time, until the ground is carpeted. I haven't seen the roots. Even after taking 2 inches of topsoil off, adding a heap of cleenfill & seeding with a tough environment grass. The grass grew beautifully. Then suddenly, before the tree started to shed, it all just died. It didn't make any difference how much I watered. It grew during winter. Getting even more sun as we came into summer had no effect either. That's when I began to wonder if there are trees that are "poisonous" (so to speak) to grass.
Yes, there are trees that are allelopathic (toxic) to other plants, including grass. Here is a good fact sheet about growing grass with trees that includes info on allelopathy towards the end. You'll notice that some kill grass. They also kill plants in different ways, including toxins in dropped leaves. http://www.extension.umn.edu/garden/landscaping/implement/trees_turf.html
It mentions that establishing grass under an already established tree can be much harder, or impossible, compared to planting a new tree into already established turf.
It is very common for new grass to grow well under trees from fall to spring and then die in summer from a combination of less water (lots of competition for water under a tree and in addition the foliage acts like a umbrella and much percipitation never reaches the ground) and less light under deciduous trees.
At any rate, your tree sounds like its dropped foliage is creating a self "mulched" area beneath it. Many trees, such as Southern magnolia and white pine, do this and these self-mulches are considered an attractive and practical way to go.
Incidentally, the "cones" on chamaecyparis don't look anything like a conventional cone. You may want to look at images in a search engine.
Even a groundcover would be hard to grow under this tree, so mulch may be the best solution.