Want this question answered?
Yes they can because they want to make sure your not a "trouble maker". I know this because some of my family members are in the military. they can may also know all of your family's name and occupation and their criminal records.
Start by checking any military records or documents your family might have. You can also reach out to the National Archives or the appropriate branch of the military for service records. Additionally, consider asking other family members or researching unit histories to see if you can identify which unit your grandfather served in.
Criminal records are a matter of Public Record and can be accessed through the individual courts where conviction was made. If the individual in question was in the custody of the state's Department of Corrections (prison), then each state maintains record of all inmates, and these records are often available on line. Military records are more private. The public may access some information, but most information is held confidential and is available only to the service member and the immediate family of deceased service members.
If your father is not available to tell you what his family history is, for whatever reason, then you have to go to other relatives who might know something or to the records. Records would include vital records (birth, death, marriage), legal records (land ownership, court records, civil cases, wills, etc.), military records, newspapers, cemetery records, etc. And, unfortunately, if you can't find something to start from, you may have to simply give up, at least for a time, and work on something else, perhaps your mother's family history.
Virtualy impossible unless it was on their medical records. You cannot find it out by using other family members blood groups either, we all get our blood group from our parents and even our grandparents.
Veterans Administration
library
The US Military Records Center, near St.Louis, MO., keeps all US military personnel records. They have a website.
Unless it was issued to the Military you can't get much and then only limited information. The Military records were researched but the early Remington records were lost in a fire.
One can find information from family pictures firstly from other members of the family who are in the picture or older members of the family, you can also get information from Genealogy site.
Ancestry's website has a list of military records. A membership is a little expensive, but you can do a fourteen day trial and it is very easy to cancel online.
Using Google you can find a lot about your family. You can also contact members of your family.