As @ColeValleyGirl says, there's a related question On what ship did Franz Joseph Kerber (1825-1873) immigrate to the US?, so let's attack the other question, "How can I find death information for Richard Turkington?"
Since you have so little information about your 2g-grandfather, my advice would be to fill out more details of your great-grandfather's life first, in the hope that you could find more clues.
- Make a research journal to record information you find, what searches you have attempted, and the results you found, both positive and negative.
- Write a biographical profile of all the people you have mentioned in your question, and an associated timeline with all the events you have found so far, and a list of all the sources you have from which that information was taken. You've already mentioned some of the negative search results where you haven't found certain sources you would expect to find, like census records. Make a wantlist of records you would like to find, in chronological order, and list the information you would hope to find from them.
- Create a research plan by examining each assertion you have, and asking how you can find other sources to confirm the information you already have. Ask also what other records you could find that share the same information with your wantlist records. You've already done some of this already (using City Directories as a substitute for residence information, because you can't find a census). What else can you find that might be a substitute for other 'missing' records?
- Use the assertions you have to formulate research questions. Try to answer some of your research questions with other questions. What happened when your great-grandfather was orphaned at the age of 13? Did he go into an orphanage? What records might exist that would talk about an event like that?
- Use your modern-day knowledge to compare and contrast while you investigate the historical period. What happens when a child is orphaned at 13 today? What agencies would handle his case? When were those agencies established? Who (if anyone) did that job before? Collect general knowledge about what the laws required in that time period.
- Research what records are available for the places and time periods you are investigating. See the related questions Where are the NYC vital records on FamilySearch? and New York State vital records collections and other questions and answers for some ideas. There are extensive pages on researching in New York City in the Family Search Wiki. See New York City New York Genealogy.
I picked out the event of your g-grandfather being orphaned because if you can find more information about out that event, you might find a clue that would lead you to more information about his father's death.
The other problem with searching for death records, or any vital records, is that there is no guarantee you will find these records in the towns that a person lived, or where their parents lived. For births, sometimes a mother traveled to where her mother lived so she could have support during and after the birth; for marriages, couples sometimes traveled to the county courthouse in the next county over to get a license, because it was easier to get there than the county seat of their own county; for deaths (sadly) people can die anywhere, suddenly, when they are away from home.
Work from what you know, and work outwards, trying to take small steps as you go instead of big leaps. If you can't find an expected record like a census record, use other sources like the City Directories that you've already collected. Any of the techniques genealogists use to fill in the gap from the loss of the 1890 census can be applied to a case where you can't find your ancestors in a census. Reading case histories is a good way to find ideas for new approaches to search -- if the person is working in the same geographical area as you, you can discover new resources to search. If not, ask "they used this resource in their area -- is there something similar for New York City / Chicago?"
Don't neglect the great resources at the New York Public Library. One place to start might be their page with Research Guides for Researching Orphans in Genealogy.
This online Guide to New York City Death Records (downloadable PDF) at FamilySearch may also be useful.