Finding a birthplace
One possible source for your great-grandfather's birthplace is in his naturalization papers. The document from 28 Aug 1886 which you list in your source list is likely to be the Petition for Naturalization ("second papers"). The US National Archives' website has an overview of the process, which takes place in two steps. NARA says:
As a general rule, naturalization was a two-step process that took a
minimum of 5 years. After residing in the United States for 2 years,
an alien could file a "declaration of intent" (so-called "first
papers") to become a citizen. After 3 additional years, the alien
could "petition for naturalization." After the petition was granted, a
certificate of citizenship was issued to the alien. These two steps
did not have to take place in the same court. As a general rule, the
"declaration of intent" generally contains more genealogically useful
information than the "petition." The "declaration" may include the
alien's month and year (or possibly the exact date) of immigration
into the United States.
If your great-grandfather arrived around 1868, and was naturalized in 1886, then the date bounds for your search for the Declaration would be around 1870 (which would satisfy the requirement to reside in the US for 2 years) and 1883 (to meet the rule of at least 3 years in between the first and second papers).
FamilySearch does have some Declarations of Intent for local courts in Illinois. These can be browsed by choosing the link for Browse All Published Collections and choosing the category "Migration and Naturalization", or by typing Naturalization in the collection title filter. Start with the Macon County courts, but if you don't find anything there, widen the search and check the courts in neighboring counties. Just like today, sometimes people visited the courthouse of the neighboring county if it happened to be closer to their home and was easier to travel to.
The FamilySearch Research Wiki's article on Illinois Naturalization and Citizenship has an overview of holding for FamilySearch and elsewhere. That article says that some naturalizations may have taken place in the Probate Courts, so there are naturalizations mixed in with the other records in FamilySearch's collection of Illinois Probate Records, 1819-1988. If you can't find your great-grandfather in any of the indexes described in that wiki article, be aware that there are often indexes to the record books in the record books themselves, at the beginning of the volume.
You may not be able to find direct evidence for your great-grandfather's birthplace, but sometimes the area a family came from in Germany and the names of the parents can be found via the records of other siblings. Sometimes other clues can be found in the records of friends, associates, and neighbors, too, because people often migrated in groups. One quick thing to check: see if the people who were witnesses on your great-grandfather's Petition for Naturalization did attestations for others nearby in the record book.
Finding his death record
For his death record, keep an eye on the Illinois Statewide Death Index, which is an ongoing project of the Illinois State Archives. Their website gives a summary of the information that the records might contain:
Death certificates show the name, age, sex, marital status, and race
of the deceased; the places of birth, death and burial; the dates of
death and burial; the cause of death; the date filed; and the
signature of the physician and the registrar.
Death record or register show the name, race, marital status, age,
sex, and occupation of the deceased; the date, place, and the primary
cause of death; contributing causes and duration; the place and date
of burial; the name and address of the undertaker; and the name and
address of the physician.
(Bear in mind that while a death certificate is supposed to show a place of birth, the information will only be as good as the informant's own knowledge. If the informant is a child or much younger relative than the deceased, it will be hearsay and not personal knowledge.)
IRAD's publication Local Governmental Records Listings, arranged by county, shows what holdings are in the Illinois Regional Archives Depositories and elsewhere.
Finding his parents
In my own research, I was able to find the maiden name of my husband's great-great grandmother by searching for the marriage and death records of all of his great-grandmother's siblings, and tracing the entire family group. If I had restricted myself to his great-grandmother's own records, I would not have been able to find the information.
- Class syllabus / handout from RootsTech2015 class RT1720: "Impossible Immigrant! Exhausting Research to find an Ancestor's Origin" by F. Warren Bittner. This PDF (available for download for a limited time) gives an outline of steps you can take to look for more information about your great-grandfather. Bittner is a meticulous researcher and a good lecturer, so I have this webinar on my "be on the lookout" list.
Class handouts from the United States Research Seminar held in October 2015 at the Family History Library:
The videos for these classes will be posted at FamilySearch after they've been cleared for viewing.
The USCIS (United States Citizenship and Immigration Service) has a Genealogy Program, from which you can request an Index Search (to see if they have records about your immigrant ancestor) and copies of the records. See their page on USCIS Webinars for the date of their next live webinar. (No recording is available, but if you can't make the live webinar, the same information is on the website -- however it's easier to understand the process when one of the USCIS staff members walks you through it.)