Q: I am looking for several death records from the second half of the nineteenth century. If one died while out of state, would it be normal for a death certificate to be filed in the home county or state? Or might it be filed in the place where he died?
Here's your rule of thumb. The death certificate will always be filed in the county/state where the person died.
Note that your question was framed on the 2nd half of the 19th Century - 1850-1900.
Depending on the year many states did not have any statewide registration of vital records - so no record would be filed in those states.
Look also for burial records. If the body of the deceased was returned to the home state for burial there will definitely be a burial certificate on file in the home state even if a death certificate is not created. Burial certificates can be interfiled with the death certificates or be kept separately. Usually the burial certificate is part of a death certificate - but depending on the local practice they might be indexed separately. So be sure to ask if they are also available.
Remember too that the census also included mortality schedules. These vary by census year and include all persons that died in the previous 12 months. Another source for you is probate records. These will be filed in the probate court of the person's home state and include the death certificate/record of the deceased.