M.L.
When she wants to force the one more hug/kiss issue, just tell her "the kids are maxed out right now, I'd rather them rest so they don't get wound up again." Or, invite her to walk to them on the couch and give the final hug there.
If you approach talking to her like it's going to be a big issue, then it will be.
But, if you gently explain in a short sentence while it's happening, then it will be just a part of the conversation.
The sippy cup thing, I would let that slide. You two have different views on it, most people I know would have done the same thing as your mom. So don't nitpick her for things like that. She can't possibly know every single rule you do or do not have. If she snapped at your son angrily, just follow up with, "Mom, it's fine he can have it, your tone was a little angry though and he picks up on those things."
As for the baby throwing food, I admit, it's weird she snapped at a 9 month old for throwing food. I would have just said calmly, "Mom, he's a baby, he doesn't really know anything else at this point, no need to get snappy."
If she is becoming increasingly sensitive, with a lot more yelling, I would calmly tell her that you love her, but are worried because she seems upset quite a lot. Maybe she is going through menopause, or having a very rough time dealing with retirement and feeling alone even more so with the death of her daughter because she isn't busy anymore.
If she snaps at your 3 year old when he is rough, then so be it. What you said here, "Also they fight all the time. I mean she just snaps really easily. Like he is rough with his brother I know this, but before he even has a chance to get his act together she is snapping at him." She is protecting a 9 month old, you don't give a 3 year old a chance to get his act together when he is being rough with a baby... you intervene immediately, which is what your mom did.
Overall, it sounds like she is an attentive and great grandmother, that doesn't like to put up with disturbances so she nips it in the bud quickly and maybe a little too harshly. Just tell her that if you need too, but choose your battles.