@Solid:
The thing is, we haven't really gotten Carrots perspective about anything. All straw hat members are protagonists and main characters in their own right, Carrot hasnt been given any of those privileges to her character, hence she is a side character (but one of the most prominent side character, because of her panel time), and I just don't see a reason for that changing, the only plot relevant thing she got going for her is Pedros death, but that might as well be something that concerns all the minks and not just Carrot alone.
As for Pedro, he said it was Neko who sent him out for the poneglyph mission, so the whole dawn thing probably stems from the dukes and their travels with Oden, they know what is needing to be done for future, and while it was personal for Pedro since he admired Roger as well, we just havn't gotten to that point with Carrot, but I think we will get there, she will be the next mink to inherit Pedros mission for gathering the poneglyphs under the Straw hat umbrella, that is the best scenario for her future imo, but we have been through this already so maybe there's no point it bringing it back.
Carrot really hasn't had the opportunity to though. However, now she does, considering how, once again, Oda constructs away for her to seemingly be present away from the rest of the Musketeers she affiliated with. Considering that she had nothing for her going into Whole Cake Island and involving her made no sense, Pedro then suddenly giving her more context on why their relationship between the Kozuki Clan and their tribe means so much more than what they know on the surface level. Pedro apparently knew, but I guess he didn't have enough time to relay it to her before blowing himself up.
Also, no, Neko did not send him on the mission. He said he simply wanted to be of service to Nekomamushi, not that Nekomamushi personally told him to go search for the Poneglyphs.
–- Update From New Post Merge ---
@Zhenja:
Does it matter what the Dawn is? Carrot tagged along without knowing anything about it.
It's very possible that Pedro also didn't know about it when he was little. And he only wanted to go with Roger because he liked him and wanted to help him.
Just as kids often do when they like/admire someone, they want their attention.
When he learned about the Dawn and that it has something to do with the Poneglyphs he figured the best way of helping would be gather information about them.
When he met the Strawhats and Luffy, he saw what kind of person he was, and therefore put his trust in him bringing the Dawn, since Roger wasn't around.
Pedro sacrificing himself means it's not necessary for him to be around to know that it gets accomplished, because of the trust he put in Luffy.
Just in case: all assumptions on my part, not trying to prove anything, just my point of view.
Yeah it does. Because if it didn't matter to her, she wouldn't have been made aware. Pedro wouldn't have discussed it with her, and normally things would play out with her just never knowing anything about it. We don't know what Pedro knew to the extent that others are aware about. He put his trust in Luffy, but then he makes Carrot aware of it. Why? Oda has been very vague about it for a reason. A lot of answers I feel we'll get on the arcs conclusion.
What Pedro did was, as he put it, lay the foundation. He didn't personally think he was the one was going to be bring forth the dawn, just that one day as was foretold, someone will need it and that he will do so in the name of both the Dukes (Chapter 878 for reference). When he knew that it would be Luffy, that literally ended the reason for any expedition group to exist after that.