As others have said, the biggest problem is that you and your husband are not on the same page. He's more social, you're more of an introvert. So some middle ground needs to be found. Do you feel that the workload is evenly distributed between the 2 of you? I'm just wondering if there is some lingering resentment that he is out socializing while you are studying, doing laundry, and getting dinner on the table. So make sure that is all worked out between the 2 of you.
Yes, the neighbor is getting mixed messages. She wants to check with you (woman to woman, which is kind of sexist in terms of women running the house) but it could be because she senses you are more hesitant. I don't know what she meant by the garage door being shut, and I think you did the right thing by calling rather than texting back due to misinterpretation. Two thoughts: 1) you don't have to answer every text immediately and you can say later, "Oh I'm sorry. I was busy with a class research project/kids in the tub/my phone was charging (pick one) and I didn't see it." 2) Next time, call her and say, "You had a question about my garage door, I gather. What's your question?" So let her explain her inquiry while you think of an answer. And it's okay to pause and let someone keep talking. Then you can answer, "I didn't even think to open the garage door as I didn't think it would bother anyone if it were shut." I wouldn't want neighbor kids in there with power tools or garden shears, and she shouldn't either. We also get squirrels and skunks roaming around and I don't want them in there either.
Next, if your husband and she are talking over the back fence, then I think it's fine for the kids to run back and forth if both adults are supervising. Leaving it unlatched all the time? No way. I wouldn't want a 2 year old in my yard if I wasn't there, and I wouldn't want my 2 kids running next door without my knowledge. So, if questioned, you can just say that you want to know where YOUR kids are and not have them make a nuisance of themselves by running around to the neighbors' houses (make it plural so it's not about her) without asking your permission. But get your husband on board with this from the get-go. Compromise - if he's there, the kids can run around with the open gate, but there's no open crossing when you're not there. You want your kids to learn to ask politely of the neighbor if her little Billy can come out and play, and that means going to her front door (or side door, whatever door she designates) just as you designate your front door. Tell your husband you don't want just anyone showing up at the glass doors if you're in your bathrobe or not in the mood. People can learn to call or ring the doorbell - which is teaching kids manners.
I'd imagine that, once her baby comes, she'll be tied up. But you want to set things straight now so that she doesn't send the 2 year old over every time the baby is napping!
If he wants to text her and vice versa and you're okay with it, fine. If he's making family decisions without consulting you, that's not okay. I had a bunch of texts with a neighbor man re sharing tools and my husband couldn't care less about being involved. But if your husband is making decisions that affect you, that's on him.
I do think you could lose a little of your edge (as in what business it is of hers if your door is closed) - you're not wrong, but make sure your disagreement with your husband isn't spilling over and becoming her problem. She doesn't know who to go to right now - although I do think you're right about her expectations of boundaries being very different than yours. If you can state your needs positively, it will make things easier. Just say you want to be prepared for visitors so you can be hospitable rather than hassled with your school projects and kids' baths. And make sure, if one family is watching the other's kids, that they really are watching them, and not just sending them home without the other parent knowing they are coming home! Make it about safety, not just protocol, and you'll do fine with her. (I hope!)
But again, start with hubby who needs to cut you some slack.