Rosie is probably gonna die soon. (almost) baseless theory
So Rosie belonged to the headmaster, yeah? We can see from the portrait of him that she was already a fully grown and trained dog before the world went to shit. As far as I'm aware it's not confirmed what breed she is but it's a reasonable assumption that the headmaster of a boarding school would have a pedigreed dog because having a non pedigreed dog is frowned upon in fancy shmancy society. Based on the assumption that Rosie is a pedigreed dog coupled with Rosie's appearance, she looks to be an American Bully.
The average lifespan of an American Bully is 8-12 years. Based on what we know of Clem and AJ's history, we're already a minimum of 8 years into the apocalypse and given that she was probably already at least 2 in the portrait with her and the headmaster, she's at least 10, and honestly she's probably older than that. She's an old lady dog and probably doesn't have much time left before her age catches up with her and she starts slowing down and deteriorating.
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Foreshadowing I guess, but other than that it seems kinda unecessary for her to die like that.
Better than getting eaten by walkers or shot/stabbed by raiders.
I guess if Clem and AJ get separated and AJ is in a bad condition, either bite or sickness, Rosie takes care of him and does what she can, but when Clem comes back she notices Rosie died and AJ is just left alone. That would be alright.
...
PREACH!!
You're overanalyzing it. I'm sure Telltale didn't pay much attention to Rosie's age. I hope they don't kill her off. One death of a dog is enough.
If Rosie those die it will be dog number 2