I won't use 产地 for animals ... not least for endangered animals (as if they were some produce we eat every day). It's a bit anthropomorphic to use 家乡 "home" here, but that sounds much better than 产地 to my ears. You can also use a verb: 有人说人类起源于非洲 "it's said that humans originated from Africa".
Understandable, but to be strict, it's incorrect.
产地 means "producing area". But 大象 and 老虎 are not productions.
Also, 印度 is not the ONLY place in where 大象 and 老虎 are living. If you want to say "is famous of producing something", consider the word 盛产。But still, only for things, not for animals.
Also incorrect. First, 发祥之地 is for things or cultures. Second, 发祥之地 means "origin". Certain spicies of 大象和老虎 have an India origin, but in fact, not all of them do.
Here are my suggestions:
印度是大象和老虎的家乡。(Using your concept of "home". It's ok because 家乡 here is just a metaphor. Although there're 大象和老虎 living in other places, a metaphor is not so logically strict comparing with 产地/发祥之地.)