Most Categories for a Single Nominee: 9
In 2013, Louie star/creator Louis CK was nominated for nine different awards, ranging from writing to performing to directing to producing to editing to hosting SNL. Keep in mind that this could not have happened until relatively recently, both because of the (relative) ease with which someone can be a metaphorical one-man band and get something on the air, but also because several technical categories weren’t even recognized for years. (The editing categories are just over 10 years old.)
That same year, Tina Fey was nominated in seven categories as a producer, writer, and lyricist (for 30 Rock's "Rural Juror" song), as well as in a special category for the sitcom's webisodes and for her role in the telecast of that year's Golden Globes. This doesn't say anything about the state of the medium -- Tina Fey is just ridiculous talented.