Gai thi= used to sing "Uski nani ne gaya"= Her grandma sang. Hope it helps. More eg. Ek larki 'jati thi'= one girl used to go, in contrary, ek larki 'gayi'= one girl went. Basically verbs with 'thi' makes it like used to go, used to sing etc.
Yes. Hindi is like latin languages such as French or Italian in this respect. In contrast to English, possessive pronouns bear the gender of what or whom is "possessed", not the person who possesses.
There are even languages like German where it bears both genders :-) sein/seine/ihr/ihre.
I did not figure out the usage of थीं. How is it different from थे and थी? Can someone help? thank you.
था : masculine singular non-honorific थी : feminine sing. non-honorific थे : masc plural or honorific थीं : fem plural or honorific
So how would you say his grandmother was singing? Like imperfect tense?
I think that "His grandmother would sing" is also a correct translation. Even though "would" is more commonly used for conditionals and future in the past, "would" can also express a habit in the past as a synonym of "used to". But maybe it is not accepted in order to avoid any confusion in a lesson about tenses?