I would talk to the tree surgeon. It will probably not be possible to take the branches down in long sections, but they are not too good for planking anyway as they tend to be twisted and knotty.
Western red cedar Thuja picata, is a fairly straight tree with small side branches, and I wouldn't think it would be grown near a house. That is the one that is usually used as a timber tree. There are other varieties of 'cedar' like eastern red cedar.
The most likely sort for a garden are the old world cedars, such as Cedar of Lebanon and Atlas Cedar.