The funeral in Arizona - I would think the family would have paid. Not sure, though.
Like Jazz said the AF-2 transport was authorized by the POTUS, but there was something that I read that said they family has to request it, because, as a Senator, he was entitled to lie in State at the US Capitol, but doesn't HAVE to. The Funeral at the National Cathedral - was definitely paid for by the US Taxpayers. The burial, which could have been at Arlington (which he didn't take advantage of - but requested Annapolis, instead) was also paid for by the US Taxpayers.
While he was certainly entitled to the military honors he received along with a burial and even the transport from AZ to DC due to his political office - I feel like it was way overblown and cost a crap-ton more money than it really should have. Especially since Ms. Cindy Loohoo can certainly have afforded all of that.
My father's and my mother's burials at Arlington - was also paid for by the US Taxpayers. We had about 1 hour before the service for my family to gather at the reception place. (Strict! 1 hour!) We had 1 car which transported my mother's cremated remains, that we followed over to the Columbarium. (a procession of about 4 cars, as there were not a lot of people attending) and a vehicle which carried the Officiant and possibly the honor guard. That was it.