Unless there's a reason to, they won't take your dog out of its kennel during a layover. (As far as I know, "a reason" would be if the next flight got delayed significantly, or if the dog appeared ill.) The airline staff won't risk losing your dog, or getting injured/having the dog get injured by taking it out of the kennel otherwise. (My cat flew NC -> NM which included a 2-hour layover and she was in her kennel the entire time.)
My dog didn't arrive with a leash (though I'm not sure whether my parents offered one to go with him either). The animal reception folk kindly gave me a leash, and I donated my crate to them, so it seemed a fair enough exchange.

I'd imagine if you asked, they could tape a leash to the kennel, though.
The paperwork should get taped to the kennel so that it's always with the dog.