First, no, no, no to walking alone. Just "no".
As far as staying home alone. If you have a neighbor next door that is a friend that you trust, you can certainly leave her in the house to run out for a short errand, under an hour. I used to do that and I'd ask my neighbor "Are you home for the next hour? Okay, I'm going to the store, the kids are alone, they know you're number. Please call 911 if you smell smoke." It worked great.
However, I don't think it's legal to leave her at home unsupervised for a longer period of time. I think in Texas kids have to be age 11, but I might be wrong.
Giving her a little freedom to be independent is a good thing, as long as she's not in danger.