Grandmother - granny. Father - daddy. Зима - зимушка. "Зимушка (родная)" ~ "dear winter". Зимушка isn't Blizzard or a very harsh and deadly winter. P.s. If I lived in the world full of zombies and I had a sword I would perhaps name it with an affectionate and soft name like "Granny" and stab zombies with it. For me it sounds although weird, but magnetic. I don't know about medieval times. P.p.s. just listened to the song, could comprehend only 5% of it x_x - The Hardcore Russian language.

no it's winter only. this is the affectionate name, usually so say in national context, such as in fairy tales, songs and so on

This is diminutive. Уменьшительно-ласкательный суффикс.

Зимушка is most frequently used as a poetical way of addressing the winter as if it were a person. Sort of 'My kind Winter'. In modern language such words are not used, because the urban society is no longer as much connected with nature as was the case with 19th century peasants.