Dear Shaun,
Your mother knows that she was not a hands on parent with you and in fact put herself, drinking and drugging before her family. No doubt she has her side of the story….the relationship between she and your father may have had elements that you are not aware of.
I suggest that you meet with your mom someplace where you can just have a heart to heart talk. Not a restaurant where she can pull her pouting thing….maybe a picnic or a walk (just the two of you). Let her know how you feel and what the ground rules will be when she is with you and your immediate family. If she can abide by your rules, then she can be a part of the family. If she can’t she won’t be included. If the latter becomes the case, I hope you will still stay in contact (one on one) and take her to lunch and shopping every now and then. You might even be able to get her into a 12 step program.
Having said this, I would not allow a child to yell at an adult, unless the child is in harm’s way. (i.e. inappropriate touching or being asked to get in a strangers car, etc.) While your mother has not gained the right to correct you, your child or animals, when your little one yelled at her, you should have corrected her (birthday or not)!
Blessings…..