There are so many conifer cultivars, that it is difficult to be certain. Even if you still had the nursery/garden-centre label, it would not be 100% reliable as a name, since mis-labelling is not that rare.
However, I think it may be a cultivar of Hinoki cypress, Chamaecyparis obtusa (the true species has a much less compact habit).
I think there is a very good chance it (the species and your particular specimen) would respond quite well to cloud/layer type pruning, as seen in many bonsai. Do note, however, that your plant, with its roots in the ground, won't grow as slowly as a bonsai. It will grow relatively slowly just by the nature of the cultivar.
Cultivars with nana in their name, such as 'Nana Gracilis', are likely to be small by nature, and probably slow growing. This example took 31 years to grow to 2m tall -
I would guess, judging by the age of the house and the adjacent maple, that yours attained its current height much quicker. So it is probably not 'Nana gracilis', but another cultivar (the RHS Plant Finder lists 113 cultivars).