Thanks for the encouragement and advice everybody!
I would love to build a nice solid wood permanent one when we own a house, but for now I thought this would be prefect as it is going to be dirt cheap and easy to move to a new place when we do. I just feel so bad that they can't go outside especially since Pooch was an outdoor cat before we got her.
We will be using chicken wire for the roof and probably for the walls as well, don't know why I used the word "mesh" instead in my OP.
I might ask my neighbor if I can use his indoor/outdoor cat (Moon, who loves me) as a "test cat" to find any weak spots. That way if she gets out, no big deal cause she is allowed out anyway!