I'm editing, just for brevity and clarity:
First: is she too young to date? (even if it's just texting and saying hi): Yes. In my family, 'dating' was reserved for at least 14/15 years old and already in high school. AND your grades had to be up. Before that age, you could meet at the roller rink or arcade with a group of friends (or other common gathering in public), but that was it.
No boyfriends or girlfriends until high school. Period. I don't believe in changing rules because 'they'll just do it anyway'...my morals and guidance don't change because of my fear my son will do it anyway. If I don't stand firm, who does he have to look up to for support and guidance--the woman who caves in?
If it were me, I'd encourage my child (daughter OR son) to focus on their interests, schoolwork, etc. and let them know that if that other person is really, truly their destiny, they're going to still be interested AND will have time to get to know each other better by hanging out at group gatherings and developing friendships. Prioritizing getting to know someone, in my opinion, teaches our kids that relationships are about more than crushes or attraction, they are about getting to know the whole person.
I would be the big bad mom and say "no". There's something to be said for having something to wait for. When I was a kid, signs at many businesses would say "12 and under, Free" or "Over 12/Adult" because 12 year olds are still perceived as children. Children do not need the complication of 'dating'. Would you also want this to complicate her friendships? It sounds like this has some potential to do so.
And no, I wouldn't be mad at grandma. Consider her generation. Consider that she's concerned about her grandchild's reputation. In my opinion, she's just telling it like it is. Yes, it would have been better if she'd saved that concern for your ears alone, but I don't think your daughter will be traumatized-- I'm sure she's heard worse spoken about other kids at school.