As I live with three dogs, I am very happy to let my dogs run around and safely play with other dogs from the neighborhood.
As what meaning can I interpret the word "as" here?
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As you have used it, As comes very close to meaning since or because but doesn't quite mean either of those things. Since or Because would be setting up a causal relationship.
If the relationship was causal and you were asked Why you let your dogs run around and play, you could say Because I have three dogs. As sets up a similar relationship with a much softer implication of causality.
As in your sentence merely establishes a relative situation without implying causality. The meaning is closer to I am a dog owner. I let my dogs run around and play. Maybe that's causal. Maybe it's not.