Is is possible or easy to milk a fallow deer, is it a quality milk, and can it be turned into cheese?
Very few animals are comfortable being milked, even many domesticated animals that have been milked for hundreds of generations are still difficult to milk. (camels, horses). Can they be milked of course it is just a pain in the ass, the animals don't like it and it is more work, you will not have factory dairy production with deer.
Any milk can be turned in to cheese, so that is not a problem. Mozzarella is actually water buffalo milk.
Historically, fallow deer have been domesticated for purposes of milking, and there is continued interest in the species to this day for producing boutique dairy products.
While deer are nervous animals, with long-term exposure to and care from humans, it is reasonable to expect that they would be no more difficult to milk than a cow (meaning that they would accept the presence of a familiar human and would allow themselves to be milked), and once accustomed to being milked, in the absence of a fawn, they would likely be no less eager to relieve the pressure and discomfort of full udders at the time that they are regularly milked than cows have been shown to be.
However, unlike a cow, the yields would be lower in both quantity per day and total number of productive days per year.