May be able to shorten. Do it in winter. Cut a little below your final height so it can grow and fill in the top. Hopefully someone will join in and tell you for sure. may need to protect the cut trunk with sealer also.
It looks like an Arborvitae(Thuja) either a 'Green Giant' or an 'Emerald Green' , though I can't be sure based on the pictures, The Green Giant can reach around 30ft tall. The Emerald Green only gets around 15 ft tall x 4-6 ft wide. Yes you can prune them. The Emerald Green is often used for topiaries, cut into spirals or pompoms Just be careful that you don't prune as such a time as new growth will be nipped by the cold, they usually push new growth out in the spring.
Hillspire Juniper
Juniperus virginiana 'Cupressifolia'