L.,
This is YOUR daughter. YOU make the decisions about how to raise her, and it is expected that those around you abide by those decisions. They may disagree, they may talk to you about why they disagree. They may not ignore your rules, or undermine your authority in front of your daughter. That is not acceptable.
As for eating issues, take your Mom to your daughter's next ped appointment, and let the Dr. tell her.
Picking up toys is a great lesson in responsibility and taking good care of one's belongings. However, at 3 1/2, she may not be ready to undertake such a large task alone. Going about it the "you put the blocks in the container, I'll pick up the books" method is completely appropriate. Its interactive, gets her helping, and you can make it fun by having a race or some other game. Most children at this age cannot follow through with a command to "clean up your toys" in a messy playroom. It's overwhelming, and they don't often have the organizational skills to look at the mess, decide which specific tasks need to be accomplished, prioritize those tasks into a logical order, and then carry through a plan to accomplish the command. That's a lot to ask of a child this age. For you to help, or to coach her through by telling her which things to pick up next is completely appropriate. But it is good for her to be an active participant in the process.
I am with you on not fighting food and toilet training. Those are two things you can FORCE a child to do. Nor should you. Manners are a big deal. How your daughter interacts with family members teaches her how to interact with the rest of the world. While Grandma may tolerate things others would consider rude, everyone else won't. It will shape how people relate to her, and that will have an effect on her self-esteem going forward. It's a big deal. Having her say hello when she meets people is important, because it forms their first impression of her and that affects how they treat her. Having her say goodbye and thank you when she leaves someone's home shows that she is polite and appreciative, and they will want her to come back again. Being liked will bolster her self-confidence, and kids need that. Is your Mom maybe uncomfortable with "Goodbye"? A lot of people, and a lot of kids, don't like that word because it sounds very final. Would your Mom be better with "see you soon" or something silly, like "later, alligator"? The two of them could have a special departure thing that's just for them, which wouldn't undermine your feeling that she needs to say "Something" in parting.
Ultimately, L., it is up to you. We raise our children, and we take responsibility for how those upbringing decisions shape who these kids become. Your Mom has done her job. Hopefully, she is proud of who you are, and confident that she did a good enough job with you when it was her turn, that she can put her faith in your ability to parent her granddaughter. Does she think she did a bad job with you? If so, then doling out parenting advice now would be pretty hypocritical. If she does think she did a good job, then she needs to be confident in you, and let you do it your way.
Best of luck,
J.