I would think that a puppy raised with the house cat
would be just fine. Dogs seem to know the difference
between a cat that belongs in thier home than
one off the street.

Even a rescue might be ok. (than you would not
have to worry about all the destructive stages
of puppy raising.)
A reputable rescue group knows all about
the personality of the dog they are placing,
and would know which of their dogs are suitable to
be places in cat home.

I have never had to reinforce a dog run,
but maybe these ideas might help.
Might be a bit expensive, but you can put in a
cement floor in, that would definitley prevent
any digging, even a partial cement floor around
all sides would work. Another idea, is to remove about a
foot of dirt, and put in a chain link floor in,
(connecting the chain link floor to the sides
of the chain link)
Than put back the dirt over the chain link floor, and put
pea gravel over on one end and grass on the other end.
I bet their are web sights that you can find on how
to build a kennel/dog run to give you ideas.