I was just in a study group talking about this sort of tree and our conifer expert discussed how difficult it is to diagnose a recently planted failure to thrive.
Assuming it was planted correctly, under-watering is a likely culprit. It is generally recommended in recent years to occasionally deeply water even established trees. (ODF reference link)
Recently planted trees need extra care for a recommended three years (link to EC 1438). This linked publication covers selecting and planting new trees, so is also useful if you need to replace this tree.
I should back up a bit: the cones are natural for this tree and something many gardeners consider an ornamental bonus. It could be that producing them when the tree is so small is a response to stress, but we don't know for sure.
The die back of oldest foliage is also natural. Each fall this will happen to some degree, known as "flagging". Your tree seems to have excessive loss, so you're right to consider changing it's care. Perhaps adequate watering next summer will generate new growth and next fall you won't be worrying. Be sure to reach in and test the soil moisture right at the root ball. One person in my study group uses moisture meters. Don't over-correct and drown the roots either. I usually use a trowel and my finger. A screw driver works too: where you hit resistance is the dry soil. It should be moist as deep as you can reach with the tool.
Lastly, one test for a newer established tree is to give it a little shove or tug. In two years the roots should be established enough that the tree is secure in the soil. There are stories about a tree rolling right out of the soil even a few years later. It looks like a "ball and socket" and can be from circling roots, too much burlap left around roots, or other reasons.
This turned into a longer answer than I commonly give, and I hope your tree survives. If you need to replant, measure the mature width of the tree you choose. The photo doesn't really show scale well, but I'm guessing this tree will be crowding the fence. The Oregon State Landscape Plants Page (link to this tree) gives the height of Chamaecyparis obtusa 'Compacta' (which isn't in the dwarf section here which surprised me). The width is only described as "broad conical" though I found other references saying 4'-15'.
Again, it's a great tree and I hope it keeps growing strong.

Many thanks for this wonderfully detailed response! So I'm pretty sure I just didn't water it enough this summer, which was its third year in the ground. I try to be water-wise, and am afraid sometimes I go overboard, trying to conserve water.