You can trim the dead wood all the way down; I usually wait until close to the end of May (or thereabouts) to see if they will leaf out. Careful when pruning "live" stuff though. These mopheads develop invisible flower buds for Spring 2018 now (in July-August 2017) so, to play it safe, try pruning anything that needs pruning by the end of June-July 2017. Otherwise, if the plant first develops the invisible flower buds and then you prune, you will not get blooms in Spring 2018. However, some of these mopheads can be re-bloomers and will develop new flower buds for a 2nd flush in the summer. Not sure if yours is one of those re-bloomers though. The prior owner might be able to tell you that and maybe also the name! Or just wait and see what happens this summer (get new blooms? if so then it is a rebloomer).

For more information, see this website:

http://hydrangeashydrangeas.com/

The shrub looks quite happy where it is now!

It is a macrophylla/big leaf hydrangea of some kind. That is what Luis gave you pruning direction for.