Two: physically he resembles Bela Lugosi’s Dracula, adding in a beard and a monocle, a symbol of bygone mainland Europe aristocracy.
Three: Why does he count? We obviously have the pun on his title of nobility. More importantly, as his song goes, he’s called the count because he really likes to count, and there’s some obscure folkloric history there. (Also note that his song is rooted in the style of Eastern European folk music.)
There is also a report of a tribe of Gypsies who believed that vampires could be kept out of the home by casting fishing nets over the doors: “Before the vampire can enter the house, he must count all the knots.”
Barber also mentions a compulsion to count poppy seeds, and that sometimes a person’s coffin would be filled with them to prevent the dead from rising. George Frederick Abbott’s Macedonian Folklore notes:
The visits of a vampire are further guarded against by scattering mustard seed over the tiles of the roof, or by barricading the door with brambles and thorn-bushes. […] The mustard […] is intended to make the Vrykolakas waste his time in counting. The same fatal weakness for arithmetic seems to best the Kalikantzari of Soutern Greece. If a sieve is handed to one, he will set to work to count the holes, as though his life depended on it. […] The Italians use a similar antidote on the Eve of St. John’s Day, when they carry about an onion-flower or a red carnation. This flower is meant for the witches, who are believed to be abroad on that evening. When it is given to them, they begin to count the petals, and long before they have accomplished this feat you are out of their reach.