Well, this is one of those uncomfortable translations. A garden is not synonymous to a yard in English, just like κήπος (garden) and αυλή (yard) are not synonymous in Greek. Αυλή might not be a garden (which is typically a place where flowers or plants grow), it might be just a school yard, or, as you've said, some sort of industrious. I see your point, but since these terms are differently used in AE and BE, we have to compromise with the one translation that corresponds to Greek best, and will cause less confusion. :/
Yes! That's completely right. You generally only need to use indefinite articles if you really need to define the quantity of something as being one. In cases where there can't be much confusion over quantity - έχω μητέρα, for example - you can just ditch them. A yard is unlikely to have two gardens, so it can be omitted here.
It's unlikely that you'd need to use indefinites with είμαι very often, as you're very unlikely to be two of something, although I guess there are some circumstances where you might need to use it.
On the other hand, you would often use them with έχω - eg έχω μία δουλειά or έχω ένα αυτοκίνητο or έχω έναν σκύλο - as you could conceivably have two or more jobs, or multiple cars, or multiple dogs.