@Rachael43
That's not okay, clearly. I am sure you have already done so, but if not, please change the code on your front door and make sure the lock on the gate is functioning properly. I know you said on this occasion, the guest was not supposed to lock the gate, but she did and then had problems unlocking it, so make sure that issue is resolved.
Then, you have two options. Either leave it alone, safe in the knowledge that the guest can no longer access your home, and mention something about it in the review/private feedback once she has gone, or you can speak to her about it politely and firmly and let her know that it's not okay for her to just let herself in your house. As her to please contact you should she have any issues with the gate in future.
As for the extra pet, I see you have an extra pet fee. Did you ask her to pay this? I feel like you might want to reword your listing a bit. You mention the extra pet fee in your extended house rules, but a lot of guests do not bother reading that detail. Meanwhile, in the main description, you mention the place is suitable for one, or maybe two, dogs. So, it's possible that she thought it didn't matter either way and there was no need to tell you.
I don't think Airbnb makes it easy with their pet fee structure, but I would clarify this on the listing to make sure that guests understand the costs for one pet as opposed to two, and also that they must let you know number of pets in advance. Perhaps that is also something you can double check on booking. Or, if you did, and the guest confirmed it was one and then simply showed up with two, that's something you needed to address on check in.