You’re probably right. Being discharged for medical reasons would seem to fall under the “general discharge” category. They also used to call them “administrative discharges”. Homosexuality used to be a crime and would have been punishable by a less desirable separation.
Yes and no. A reduction in pay grade will result in the reduction of rank, but a pay grade is not the same thing as a rank. You can have soldiers of the same pay grade who are of different ranks (e.g., an Army corporal and specialist). In this case, they are of the same pay grade (E-4), but the corporal is of higher rank than the specialist. In the Air Force, I believe, a Chief Master Sergeant with First Sergeant Status is of higher rank than a Chief Master Sergeant, even though the former actually has a lower pay grade (E-8) than the latter (E-9).
Bottom line, pay grade does not equal rank.