Half siblings other parent

My dad was engaged a little before he met my mom and my sister was conceived in that relationship. Ill call my sister T and her mom C. My dad and C broke off the engagement right before finding out they were pregnant and decided to stay separated and co-parent.

My dad met my mom shortly after and had my brothers and I. My parents have been married ever since. T has always lived with her mom, but we were all pretty close in age and we saw her several times a week, she came to every holiday and vacation, my mom's side of the family has always treated her like family.

My mom and C always got along pretty well and I've known C my entire life. My mom is Ts step mom bc she is married to her dad. But Ts mom is not my step mom.

So my questions are, is there a word for what C would be considered to me? She's not a mom or an aunt or even step mom but I've known her my entire life and consider her to be like distant family I guess?

Am I expected to invite my sisters mom to my wedding? Is it rude not to? Is it weird if I do?

How much DNA would I share with my sisters kids and vica versa?

TIA